Theory and Applications of Categories, Vol. 32, No. 45, 2017, pp. 1563–1587.

STABILITY PROPERTIES CHARACTERISING n-PERMUTABLE CATEGORIES

PIERRE-ALAIN JACQMIN, DIANA RODELO

Abstract. The purpose of this paper is two-fold. A first and more concrete aim is to characterise n-permutable categories through certain stability properties of regular epimorphisms. These characterisations allow us to recover the ternary terms and the n 1 -ary terms describing n-permutable varieties of universal algebras.

A second and more abstract aim is to explain two proof techniques, by using the above characterisation as an opportunity to provide explicit new examples of their use:

- an embedding theorem for n-permutable categories which allows us to follow the varietal proof to show that an n-permutable category has certain properties;

- the theory of unconditional exactness properties which allows us to remove the assumption of the existence of colimits, in particular when we use the approximate co-operations approach to show that a regular category is n-permutable.

Introduction

A variety of universal algebras is called n-permutable, n 2, when any pair of congruences R and S on a same algebra n-permutes: R, S _{n} S, R _{n}, where R, S _{n} denotes the composite RSRS of R and S, n times. This notion determines a sequence of families of varieties, whose first two instances are well known: for n 2, we regain 2-permutable varieties [23], better known as Mal’tsev varieties; for n 3, these are the 3-permutable varieties. The property of n-permutability makes sense in any regular category and was generalised to this categorical context in [7], where n-permutable categories were first studied.

A variety is a Mal’tsev variety precisely when its theory contains a ternary operation p such that the identities p x, y, y x and p x, x, y y hold [23]. 3-permutable varieties are characterised by the existence of two ternary operations r and s satisfying the identities r x, y, y x, r x, x, y s x, y, y and s x, x, y y [13]. Equivalently, they are characterised by the existence of quaternary operations p and q such that the

Financial support from FNRS grant 1.A741.14 is gratefully acknowledged by the first author. The second author acknowledges partial financial assistance by Centro de Matemática da Universidade de Coimbra—UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

Received by the editors 2017-05-15 and, in final form, 2017-12-18. Transmitted by Walter Tholen. Published on 2017-12-19. 2010 Mathematics Subject Classification: 08A30, 08B05, 08C05, 18B15, 18B99, 18C99, 18A32. Key words and phrases: Mal’tsev category, Goursat category, n-permutable category, embedding

theorem, unconditional exactness property.

\begin{matrix}c \\ ◯ \end{matrix} Pierre-Alain Jacqmin, Diana Rodelo, 2017. Permission to copy for private use granted.

identities p x, y, y, z x, p x, x, y, y q x, x, y, y and q x, y, y, z z hold [13]. For an n-permutable variety, its theory contains n 1 ternary operations or, equivalently, n 1 n 1 -ary operations where similar generalised identities hold ([13], see Theorems 1.1 and 1.2 below).

It was shown in [10, 12] that 3-permutable categories (also called Goursat categories [8, 7]) can be characterised by certain stability properties for regular epimorphisms. The stability property from [12] (which is recalled here as Proposition 2.2), visualised by means of a cube, allows us to recover the ternary terms r and s characterising 3-permutable varieties. The characterisation from [10], stated through the notion of Goursat pushout, allows us to obtain the quaternary varietal terms p and q.

Purpose of the paper. Our first aim with this paper is to extend the above characterisations of Goursat categories from [10, 12] to the general case of n-permutable categories. The idea is to obtain a property involving a diagram which allows us to recover the varietal terms describing n-permutable varieties of universal algebras. We first prove that a certain stability property of regular epimorphisms characterises n-permutable categories for n 3 (Theorem 3.3); and this allows us to recover the ternary terms (Section 4). Then, we prove a similar characterisation from which we can recover the n 1 -ary terms. For this second result, we have to distinguish four cases, depending on the congruence class modulo 4 of n (Theorems 5.1, 5.3, 5.4 and 5.5).

The proofs we give of these characterisations require two special techniques which were developed in [16, 17, 14]. A second main aim of this paper is to use the above characterisations as an opportunity to give concrete and new examples of where and how such techniques may be applied.

First proof technique: translating varietal proofs. The first technique arose from a wider historical context. Since the 1990’s, with the introduction of Mal’tsev [9], Goursat and protomodular [4] categories, many non-abelian categorical-algebraic properties have been studied. Several of those, when considered in a varietal context, led to characterisations given in terms of operations and identities. In some cases, the terms and axioms are quite simple to express: for instance, as recalled above, this happens with 2-permutability of congruences. Eventually, such a translation process leads to varietal proofs which can be sometimes easier than the corresponding categorical ones. A question thus arises:

How can one turn a (simple) varietal proof using operations and identities into a categorical proof?

STABILITY PROPERTIES CHARACTERISING n-PERMUTABLE CATEGORIES 1565

In [19], Z. Janelidze proposed a matrix presentation of some of these categorical conditions in order to study them in a unified way. Such a presentation applies, for instance, to the collections of Mal’tsev, unital [5], strongly unital [5] and subtractive [18] categories. With the more general type of matrices from [20], one can also include npermutable categories as examples. Note that these are collections of categories whose varietal counterpart has a simple characterisation in terms of operations and identities; their “categorical version” may be described in terms of properties of (internal) relations. For each categorical condition, the matrix is built according to the corresponding varietal operations and identities. At the same time, the varietal information put into the matrix allows us to deduce properties concerning relations, which translates into the categorical condition in question. Thus, the matrix gives a link between the varietal and categorical worlds. In particular, the matrix presentation makes it possible to turn varietal proofs into categorical ones: we write the varietal proof into the appropriate matrix system to get the right (but sometimes unintuitive) relation to which we apply the assumptions on the category. This technique has been explained and used in [22].

In parallel with the development of the matrix approach, D. Bourn and Z. Janelidze showed in [6] that in regular Mal’tsev categories with binary coproducts, the objects can be equipped with an approximate Mal’tsev co-operation. Using this co-operation, many varietal proofs for Mal’tsev varieties can be extended to the context of regular Mal’tsev categories with binary coproducts. This result has been generalised for Janelidze’s simple matrix conditions in [21], for n-permutable categories in [24] and for Janelidze’s generalised matrix conditions in [16, 14].

On the one hand, the matrix technique is sometimes unintuitive and may give rise to rather long proofs (see Remark 2.4). Moreover, the link between the initial varietal proof and the deduced categorical proof is often hard to see. On the other hand, with the technique of approximate co-operations, the categorical proof obtained is a simple adjustment of the original varietal proof. However, the additional assumption on the existence of binary coproducts is unwelcome. To overcome these flaws, in [15, 16, 14] new embedding theorems have been developed for Janelidze’s generalised matrix conditions.

As a consequence of one of these embedding theorems, in an n-permutable context we now only have to produce a proof of theorems about finite limits and regular epimorphisms in a particular essentially algebraic (i.e., locally presentable) category in order to obtain the same result in any n-permutable regular categories. Moreover, as we shall see, it seems that transforming a varietal proof into a proof for this particular category is usually straightforward. In each of our characterising theorems, we will use this embedding theorem to prove one implication of our characterisations.

Second proof technique: removing colimit assumptions. The second technique, from [17, 14], is used in this paper to prove the converse implications. We first show these implications under the additional condition that the category has binary coproducts, so that we can apply the characterisation of n-permutable categories with binary coproducts via approximate Hagemann-Mitschke co-operations [24]. Then, in order to remove this assumption on coproducts, we are going to use the theory of unconditional exactness properties [17, 14]. This is a class of exactness properties, stable under the cofiltered limit completion C Lex C, Set ^{op}, where Lex C, Set is the category of left exact functors from C to the category Set of sets. We thus have to check that, in a regular context, the characterising properties in our theorems are unconditional exactness properties. To achieve this, we need to check, among other things, that certain categories generated by some finite conditional graphs are finite. We develop an algorithm to do this easily.

Once this check is done, we know that if a regular category satisfies the property under consideration, then so does the cocomplete category Lex C, Set ^{op}. Using the first part of the proof, we will be able to conclude that it is n-permutable and, hence, that so is C, since the full embedding C Lex C, Set ^{op} preserves finite limits and colimits.

Structure of the paper. The paper is organised as follows. In the first section, we recall the characterisations of n-permutable varieties, the corresponding embedding theorems and prove (recall) some characterisations of n-permutable categories using the calculus of relations (Theorem 1.3). In the second section, we characterise those categories via a stability property for regular epimorphisms, using binary coproducts (Theorem 2.3). In Section 3, we prove the assumption on coproducts may be removed from this theorem using unconditional exactness properties (Theorem 3.3). Section 4 uses this result to extract the ternary operations for n-permutable varieties. Finally, the last section is devoted to the other characterisation of n-permutable categories (Theorems 5.1, 5.3, 5.4 and 5.5) which allows one to better understand the existence of the suitable n 1 -ary operations in the varietal case from a categorical viewpoint.

Acknowledgements

The authors would like to warmly thank Tim Van der Linden for his useful comments to improve the introduction of this paper and make the body of the text easier to read.

1. n-permutable categories

A variety of universal algebras has n-permutable congruences when any pair of congruences R and S on a same algebra is such that

R, S _{n} S, R _{n},

where R, S _{n} denotes the composite RSRS , of n alternating factors. This property was studied in a categorical context in [7] and led to the notion of n-permutable category, defined as a regular category [1] whose equivalence relations are n-permutable. We recall that 2-permutable categories are usually called regular Mal’tsev categories [8] and 3permutable categories are also known as Goursat categories [8, 7].

It is well known that a Mal’tsev variety V of universal algebras is such that its theory admits a ternary operation p satisfying p x, y, y x and p x, x, y y [23]. For the strictly weaker notion of a 3-permutable variety, the corresponding theory admits ternary operations r and s satisfying r x, y, y x, r x, x, y s x, y, y and s x, x, y y [13]. In general, n-permutable varieties can be characterised by the existence of ternary operations satisfying suitable identities:

1.1. Theorem. (Theorem 2 of [13]) Let n 2 and let V be a variety of universal algebras. The following statements are equivalent:

(i) the congruence relations of every algebra of V are n-permutable;

(ii) there exist ternary algebraic operations w_{1}, …, w_{n 1} of the algebraic theory of V for which the identities

 ;  ;  ;  ; \begin{matrix}w_{1} x, y, y x, \\ w_{i} x, x, y w_{i 1} x, y, y , & for i 1, …, n 2 , \\ w_{n 1} x, x, y y \end{matrix}

hold.

In order to prove the stability properties characterising n-permutable categories, we will need the embedding theorem from [16, 14]. The objects of the essentially algebraic n-permutable category Mod Γ_{n} are S_{n}-sorted sets A_{s} _{s S_{n}} (for a fixed set S_{n} of sorts) equipped with, among others, for each s S_{n}, an injective operation α^{s}: s s, 0 where s, 0 is another sort in S_{n}, n 1 ternary operations w_{1}^{s}, …, w_{n 1}^{s} : s^{3} s, 0 and a partial operation π^{s}: s, 0 s defined exactly on the image of α^{s} such that the identities

 ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}w_{1}^{s} x, y, y α^{s} x , \\ w_{i}^{s} x, x, y w_{i 1}^{s} x, y, y , & for i 1, …, n 2 , \\ w_{n 1}^{s} x, x, y α^{s} y , \\ π^{s} α^{s} x x \end{matrix}  (1)

hold for each x, y A_{s}. Roughly speaking, the embedding theorem then says that to prove a theorem about finite limits and regular epimorphisms in an n-permutable context, it is enough to prove it in Mod Γ_{n} , supposing that the given regular epimorphisms are actually S_{n}-sorted functions which are surjective in each sort.

It is well-known that n-permutable varieties may also be characterised by the existence of n 1 -ary operations:

1.2. Theorem. (Theorem 1 of [13]) Let n 2 and let V be a variety of universal algebras. The following statements are equivalent:

(i) the congruence relations of every algebra of V are n-permutable;

(ii) there exist n 1 -ary algebraic operations v_{0}, v_{1}, …, v_{n 1} , v_{n} of the algebraic theory of V for which the identities

 ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}v_{0} x_{0}, …, x_{n} x_{0}, \\ v_{i 1} x_{0}, x_{0}, x_{2}, x_{2}, … v_{i} x_{0}, x_{0}, x_{2}, x_{2}, …, & i even, \\ v_{i 1} x_{0}, x_{1}, x_{1}, x_{3}, x_{3}, … v_{i} x_{0}, x_{1}, x_{1}, x_{3}, x_{3}, …, & i odd, \\ v_{n} x_{0}, …, x_{n} x_{n} \end{matrix}

hold.

Since v_{0} and v_{n} are just projections, one usually simply refers to the main n 1 terms v_{1}, …, v_{n 1} , where v_{1} x_{0}, x_{1}, x_{1}, x_{2}, x_{2}, … x_{0} replaces the first and third identities, when i 1, and v_{n 1} …, x_{j 1} , x_{j 1} , x_{j} x_{j} replaces the last and the second or third identities, when i n even or odd, respectively.

Analogously to the ternary operation case, we also have an embedding theorem for n-permutable categories using the n 1 -ary operations [16, 14]. This time, we use the essentially algebraic n-permutable category Mod Γ_{n}^{}. Its objects are S_{n}^{}-sorted sets A_{s} _{s S_{n}^{}}(for a fixed set S_{n}^{}of sorts) equipped with, among others, for each s S_{n}^{}, an injective operation α^{s}: s s, 0 where s, 0 is another sort in S_{n}^{}, n 1 -ary operations v_{0}^{s}, …, v_{n}^{s}: s^{n 1} s, 0 and a partial operation π^{s}: s, 0 s defined exactly on the image of α^{s} such that the identities

 ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ;  ; \begin{matrix}v_{0}^{s} x_{0}, …, x_{n} α^{s} x_{0} , \\ v_{i 1}^{s} x_{0}, x_{0}, x_{2}, x_{2}, … v_{i}^{s} x_{0}, x_{0}, x_{2}, x_{2}, …, & i even, \\ v_{i 1}^{s} x_{0}, x_{1}, x_{1}, x_{3}, x_{3}, … v_{i}^{s} x_{0}, x_{1}, x_{1}, x_{3}, x_{3}, …, & i odd, \\ v_{n}^{s} x_{0}, …, x_{n} α^{s} x_{n} , \\ π^{s} α^{s} x x \end{matrix}  (2)

hold for x, x_{0}, x_{1}, …, x_{n} A_{s}. Roughly speaking, the embedding theorem then says that to prove a theorem about finite limits and regular epimorphisms in an n-permutable context, it is enough to prove it in Mod Γ_{n}^{}, supposing that the given regular epimorphisms are actually S_{n}^{}-sorted functions which are surjective in each sort.

We conclude this section with a result that gathers some known characterisations of n-permutable categories as well as a new one, given by condition (iv), which will be useful in the next section.

1.3. Theorem. Let n 2 and let C be a regular category. The following statements are equivalent:

(i) C is an n-permutable category;

(ii) P, P^{}_{n 1} P, P^{}_{n 1} , for any relation P;

(iii) E^{}E^{n 1} , for any reflexive relation E;

(iv) 1_{X} T T^{}1_{X} T T^{n 1} , for any relation T on an object X.

Proof. (i) (ii). By Theorem 3.5 of [7].

(i) (iii). By Theorem 1 in [22].

(i) (iv). This type of equivalence was mentioned at the end of [22] (and also in Theorem 1.1 in [12] for the case n 3). Note that condition (iv) is a stronger version of condition (iii), so we only have to prove that (i) (iv); in fact, we shall prove that (ii) (iv). In a regular context, it suffices to give a proof in set-theoretical terms due to Barr’s embedding theorem [2]. Suppose that xT x, yT x and yT y. We want to prove that xT^{n 1} y, i.e., that xT μ_{1}T μ_{2} μ_{n 3} T μ_{n 2} T y, for some μ_{1}, …, μ_{n 2} X. Case n 2k 1 is odd: We define a relation P from X^{k} to X^{k} by: a_{1}, a_{2}, …, a_{k 1} , a_{k} P b_{1}, b_{2}, …, b_{k 1} , b_{k} if and only if

\begin{matrix}a_{i}T b_{i}, & 1 i k \\ b_{j}T a_{j 1} , & 1 j k 1. \end{matrix}

From

x, …, x P x, …, x
y, x, …, x P x, …, x
y, x, …, x P y, x, …, x
y, y, x, …, x P y, x, …, x
⋮
y, … y, x P y, …, y, x
y, …, y P y, …, y, x ,

we get x, …, x P, P^{}_{2k n 1} y, …, y . By assumption, we conclude that x, …, x P, P^{}_{n 1} y, …, y , i.e.,

x, …, x P ν_{11}, …, ν_{1k}
ν_{21}, …, ν_{2k} P ν_{11}, …, ν_{1k}
ν_{21}, …, ν_{2k} P ν_{31}, …, ν_{3k}
ν_{41}, …, ν_{4k} P ν_{31}, …, ν_{3k}
⋮
ν_{n 3,1} , …, ν_{n 3,k} P ν_{n 2,1} , …, ν_{n 2,k}
y, …, y P ν_{n 2,1} , …, ν_{n 2,k} ,

for some ν_{i1}, …, ν_{ik} X^{k}, 1 i n 2. From these relations, we get

xT ν_{11}, ν_{11}T ν_{22}, ν_{22}T ν_{32}, ν_{32}T ν_{43}, …, ν_{n 3,k 1} T ν_{n 2,k 1} and ν_{n 2,k 1} T y,

which implies that xT^{n 1} y. Case n 2k is even: We define a relation P similarly as in the odd case. We have the same relations on P as above, with an extra relation

y, …, y P y, …, y ,

so that x, …, x P, P^{}_{2k 1 n 1} y, …, y . By assumption, we conclude that x, …, x P, P^{}_{n 1} y, …, y . We then get similar relations for P as in the odd case, but with y, …, y ending on the right in the last relation

x, …, x P ν_{11}, …, ν_{1k}
ν_{21}, …, ν_{2k} P ν_{11}, …, ν_{1k}
ν_{21}, …, ν_{2k} P ν_{31}, …, ν_{3k}
ν_{41}, …, ν_{4k} P ν_{31}, …, ν_{3k}
⋮
ν_{n 2,1} , …, ν_{n 2,k} P ν_{n 3,1} , …, ν_{n 3,k}
ν_{n 2,1} , …, ν_{n 2,k} P y, …, y .

From these relations, we get

xT ν_{11}, ν_{11}T ν_{22}, ν_{22}T ν_{32}, ν_{32}T ν_{43}, …, ν_{n 3,k 1} T ν_{n 2,k} and ν_{n 2,k} T y,

which implies that xT^{n 1} y.

1.4. Remark. The proof of Theorem 1.3 (ii) (iv) followed the matrix technique mentioned in the introduction. A varietal proof of this property was put into an appropriate matrix, from which we deduced the relation P. Although P carries the precise information we need, the varietal proof behind its construction is not visible at first glance. In the next sections, we rather give the main proofs using embedding theorems for n-permutable categories, which are mere adjustments of the varietal proofs.

2. Stability property for regular epimorphisms

In this section we are going to study a certain type of stability property for regular epimorphisms (in a regular category) which characterises n-permutable categories. The idea is to obtain, for arbitrary n 3, similar properties as those known for the Mal’tsev and the Goursat cases.

For Mal’tsev categories, the stability property involves a commutative (upward and downward) cube where the left and right faces are pullbacks of split epimorphisms along arbitrary morphisms

\xymatrix{X _{Y} Z \ar@{=>}[dd] \ar@{=>}[dr] \ar@{=>}[rr]^{λ} &  & U _{V} W \ar@{=>}[dd] \ar@{=>}[dr] \\  & Z \ar@{=>}[dd]_{g} \ar@{=>}[rr]^{δ} &  & W \ar@{=>}[dd]_{k} \\ X \ar@{=>}[uu] \ar@{=>}[dr]_{f} \ar@{=>}[rr]^{γ} &  & U \ar@{=>}[uu] \ar@{=>}[dr]_{h} \\  & Y \ar@{=>}[uu] \ar@{=>}[rr]_{β} &  & V. \ar@{=>}[uu]}  (3)

In (3), the upward morphisms are splittings of the downward morphisms, λ is induced by the universal property of the right-hand pullback and the “obvious” squares commute.

2.1. Proposition. [11] Let C be a regular category. The following statements are equivalent:

(i) C is a Mal’tsev category;

(ii) for any diagram (3), if γ, β and δ are regular epimorphisms, then the comparison morphism λ is also a regular epimorphism.

For the weaker notion of a Goursat category, the stability property now involves a cube where the left and right faces are pullbacks of split epimorphisms and where β, γ, δ and λ commute with the split epimorphisms and with their sections

\xymatrix{X _{Y} Z \ar@{=>}[dd] \ar@{=>}[dr] \ar@{=>}[rr]^{λ} &  & U _{V} W \ar@{=>}[dd] \ar@{=>}[dr] \\  & Z \ar@{=>}[dd]_{g} \ar@{=>}[rr]^{δ} \ar@{=>}[ul] &  & W \ar@{=>}[dd]_{k} \ar@{=>}[ul] \\ X \ar@{=>}[uu] \ar@{=>}[dr]_{f} \ar@{=>}[rr]^{γ} &  & U \ar@{=>}[uu] \ar@{=>}[dr]_{h} \\  & Y \ar@{=>}[uu] \ar@{=>}[rr]_{β} \ar@{=>}[ul] &  & V. \ar@{=>}[uu] \ar@{=>}[ul]}  (4)

In (4), the downward and the frontward morphisms are split epimorphisms together with a chosen section, λ is induced by the universal property of the right-hand pullback and all “obvious” squares commute.

2.2. Proposition. [12] Let C be a regular category. The following statements are equivalent:

(i) C is a Goursat category;

(ii) for any diagram (4), if γ, β and δ are regular epimorphisms, then the comparison morphism λ is also a regular epimorphism.

We now give a stability property for regular epimorphisms which shall characterise n-permutable categories for n 3. Consider the following diagram

[figure: L; π_{1}; π_{2}; π_{3}; π_{n 2}; π_{n 1}; λ; U _{V} W; X_{1}; γ; f_{1}; U; h; X_{2}; f_{2}; g_{1}; Y_{1}; s_{1}; t_{1}; β_{1}; X_{3}; g_{2}; ; Y_{2}; s_{2}; t_{2}; β_{2}; ; X_{n 2}; f_{n 2}; X_{n 1}; g_{n 2}; δ; W; k; Y_{n 2}; s_{n 2}; t_{n 2}; β_{n 2}; ⋮; V; u; w] (5)

where the left side is a limit of split epimorphisms and the right quadrangle is a pullback of split epimorphisms. We assume that the arrows γ, β_{1} and δ, β_{n 2} commute with the corresponding split epimorphisms as well as with their splittings. We also assume that β_{j} is a split epimorphism, split by i_{j}, for 2 j n 2, and are such that the following squares commute (also with the vertical splittings),

\xymatrix{X_{j} \ar@{=>}[d]_{g_{j 1}} \ar@{=>}[r]^{f_{j}} & Y_{j} \ar@{=>}[d]_{β_{j}} \\ Y_{j 1} \ar@{=>}[u]_{t_{j 1}} \ar@{=>}[r]_{β_{j 1}} & V \ar@{=>}[u]_{i_{j}}}

i.e.,

\begin{aligned}β_{j}f_{j} β_{j 1} g_{j 1} \\ & , 2 j n 2. \\ f_{j}t_{j 1} i_{j}β_{j 1} \end{aligned}  (6)

We will also consider the above diagram in the case when f_{1} f_{n 2} f,

g_{1}  g_{n 2} g, β_{1}  β_{n 2} β

[figure: L; π_{1}; π_{2}; π_{3}; π_{n 2}; π_{n 1}; λ; U _{V} W; X; γ; f; U; h; X; f; g; Y; s; t; β; X; g; ; Y; s; t; β; ; X; f; X; g; δ; W; k; Y; s; t; β; ⋮; V; u; w] (7)

and β is endowed with two splittings i and i^{}so that now

 ;  ; \begin{matrix}βf βg \\ ft iβ \\ gs i^{}β. \end{matrix}  (8)

2.3. Theorem. Let n 3 and let C be a regular category with binary coproducts. The following statements are equivalent:

(i) C is an n-permutable category;

(ii) if γ and δ in diagram (5) are regular epimorphisms, then the comparison morphism λ is also a regular epimorphism;

(iii) if γ and δ in diagram (7) are regular epimorphisms, then the comparison morphism λ is also a regular epimorphism.

Proof. (i) (ii) Using the embedding theorem for n-permutable categories, it is enough to prove this implication in Mod Γ_{n} . So let s S_{n} and a, b U _{V} W _{s}. Since regular epimorphisms are supposed to be surjective in each sort, there exist x_{1} X_{1} _{s} and x_{n 1} X_{n 1} _{s} such that γ x_{1} a and δ x_{n 1} b. Since h a k b , we get

β_{1}f_{1} x_{1} hγ x_{1} kδ x_{n 1} β_{n 2} g_{n 2} x_{n 1} .  (9)

Using the identities of (6) repeatedly, for 1 j n 2 we see that

β_{j}g_{j}s_{j 1} g_{j 1} s_{n 2} g_{n 2} x_{n 1} β_{n 2} g_{n 2} x_{n 1}  (10)

(for j n 2, just write β_{n 2} g_{n 2} x_{n 1} on the left) and

β_{j}f_{j}t_{j 1} f_{j 1} t_{1}f_{1} x_{1} β_{1}f_{1} x_{1}  (11)

(for j 1, just write β_{1}f_{1} x_{1} on the left). We then deduce

γs_{1}g_{1} s_{n 2} g_{n 2} x_{n 1} ^{(10)(9)} γs_{1}f_{1} x_{1}  (12)

and

δt_{n 2} f_{n 2} t_{1}f_{1} x_{1} ^{(11)(9)} δt_{n 2} g_{n 2} x_{n 1} .  (13)

For 2 j n 2, we also find

\begin{aligned}f_{j}t_{j 1} g_{j 1} s_{j}g_{j} s_{n 2} g_{n 2} x_{n 1} & ^{6} & i_{j}β_{j 1} g_{j 1} s_{j}g_{j} s_{n 2} g_{n 2} x_{n 1} \\ & ^{10} & i_{j}β_{n 2} g_{n 2} x_{n 1} \\ & ^{9} & i_{j}β_{1}f_{1} x_{1} \\ & ^{11} & i_{j}β_{j 1} f_{j 1} t_{j 2} f_{j 2} t_{1}f_{1} x_{1} \\ & ^{6} & f_{j}t_{j 1} f_{j 1} t_{j 2} f_{j 2} t_{1}f_{1} x_{1} . \end{aligned}  (14)

Now, we define

\overline{x_{1}} w_{1}^{s} x_{1}, s_{1}f_{1} x_{1} , s_{1}g_{1} s_{n 2} g_{n 2} x_{n 1} X_{1} _{s,0}

and for 2 j n 1,

\overline{x_{j}} w_{j}^{s} t_{j 1} f_{j 1} t_{1}f_{1} x_{1} , t_{j 1} g_{j 1} s_{j}g_{j} s_{n 2} g_{n 2} x_{n 1} , s_{j}g_{j} s_{n 2} g_{n 2} x_{n 1} X_{j} _{s,0}.

We can compute the following identities:

f_{1} \overline{x_{1}}  w_{1}^{s} f_{1} x_{1} , f_{1}s_{1}f_{1} x_{1} , f_{1}s_{1}g_{1} s_{n 2} g_{n 2} x_{n 1}
w_{1}^{s} f_{1} x_{1} , f_{1} x_{1} , g_{1}s_{2} s_{n 2} g_{n 2} x_{n 1}
^{1} w_{2}^{s} f_{1} x_{1} , g_{1}s_{2} s_{n 2} g_{n 2} x_{n 1} , g_{1}s_{2} s_{n 2} g_{n 2} x_{n 1}
w_{2}^{s} g_{1}t_{1}f_{1} x_{1} , g_{1}t_{1}g_{1}s_{2} s_{n 2} g_{n 2} x_{n 1} , g_{1}s_{2} s_{n 2} g_{n 2} x_{n 1}
g_{1} \overline{x_{2}}

and for each 2 j n 2,

f_{j} \overline{x_{j}} w_{j}^{s} f_{j}t_{j 1} f_{j 1} t_{1}f_{1} x_{1} , f_{j}t_{j 1} g_{j 1} s_{j}g_{j} s_{n 2} g_{n 2} x_{n 1} , f_{j}s_{j}g_{j} s_{n 2} g_{n 2} x_{n 1}

\begin{matrix}^{14} & w_{j}^{s} f_{j}t_{j 1} f_{j 1} t_{1}f_{1} x_{1} , f_{j}t_{j 1} f_{j 1} t_{1}f_{1} x_{1} , g_{j} s_{n 2} g_{n 2} x_{n 1} \\ ^{1} & w_{j 1}^{s} f_{j}t_{j 1} f_{j 1} t_{1}f_{1} x_{1} , g_{j} s_{n 2} g_{n 2} x_{n 1} , g_{j} s_{n 2} g_{n 2} x_{n 1} \\ & w_{j 1}^{s} g_{j}t_{j}f_{j} t_{1}f_{1} x_{1} , g_{j}s_{j 1} s_{n 2} g_{n 2} x_{n 1} , g_{j}s_{j 1} s_{n 2} g_{n 2} x_{n 1} \\ & g_{j} \overline{x_{j 1}} . \end{matrix}

This exactly means that \overline{x_{1}}, …, \overline{x_{n 1}} L_{s,0}. Moreover, the equalities

γ \overline{x_{1}}  w_{1}^{s} γ x_{1} , γs_{1}f_{1} x_{1} , γs_{1}g_{1} s_{n 2} g_{n 2} x_{n 1}
^{12} w_{1}^{s} γ x_{1} , γs_{1}f_{1} x_{1} , γs_{1}f_{1} x_{1}
^{1} α^{s} γ x_{1}
α^{s} a

and

δ \overline{x_{n 1}}  w_{n 1}^{s} δt_{n 2} f_{n 2} t_{1}f_{1} x_{1} , δt_{n 2} g_{n 2} x_{n 1} , δ x_{n 1}
^{13} w_{n 1}^{s} δt_{n 2} g_{n 2} x_{n 1} , δt_{n 2} g_{n 2} x_{n 1} , δ x_{n 1}
^{1} α^{s} δ x_{n 1}
α^{s} b

hold. This implies that

λ \overline{x_{1}}, …, \overline{x_{n 1}} α^{s} a , α^{s} b α^{s} a, b

and

a, b ^{1} π^{s} α^{s} a, b π^{s} λ \overline{x_{1}}, …, \overline{x_{n 1}} Im λ _{s}

which proves that λ is a regular epimorphism.

(ii) (iii) Obvious.

(iii) (i) By Theorem 4 from [24], we must show that, for any object X in C, the morphism ξ_{X} in the dotted limit of the outer diagram of solid arrows

[figure: X; ι_{1}; ι_{2}; 2X; 2X; A X; ξ_{X}; _{w_{1} _{X}}; _{w_{2} _{X}}; _{w_{3} _{X}}; _{w_{n 2} _{X}}; _{w_{n 1} _{X}}; 3X; 1_{X} ∇_{X}; ∇_{X} 1_{X}; 3X; ∇_{X} 1_{X}; 1_{X} ∇_{X}; 2X; 3X; 1_{X} ∇_{X}; 3X; ∇_{X} 1_{X}; 2X; 2X; 3X; ](15)

is a regular epimorphism.

We consider diagram (7), where f δ ∇_{X} 1_{X}, s ι_{2} ι_{3} , g γ 1_{X} ∇_{X}, t ι_{1} ι_{2} , β k h ∇_{X}, w i ι_{1} and u i^{}ι_{2}. By assumption, λ is a regular epimorphism.

We now consider the morphism j induced by the universal property of the kernel pair of ∇_{X}

[figure: X; ι_{1}; ι_{2}; j; Eq ∇_{X}; ρ_{2}; ρ_{1}; 2X; ∇_{X}; 2X; ∇_{X}; X]

and its pullback along the regular epimorphism λ

[figure: Q; λ^{}; j^{}; X; j; L; λ; Eq ∇_{X} .]

We get a cone for the outer diagram in (15):

[figure: X; ι_{1}; ι_{2}; 2X; 2X; Q; λ^{}; π_{1}j^{}; π_{2}j^{}; π_{3}j^{}; π_{n 2} j^{}; π_{n 1} j^{}; 3X; 1_{X} ∇_{X}; ∇_{X} 1_{X}; 3X; ∇_{X} 1_{X}; 1_{X} ∇_{X}; 2X; 3X; 1_{X} ∇_{X}; 3X; ∇_{X} 1_{X}; 2X; 2X; 3X; ]

As a consequence, there exists a unique τ: Q A X such that, in particular, ξ_{X}τ λ^{}. Finally, ξ_{X} is a regular epimorphism since so is λ^{}.

2.4. Remark. Instead of using the embedding theorem for n-permutable categories to prove (i) (ii) of Theorem 2.3, one could have used the matrix presentation from [20] and Barr’s embedding theorem for regular categories [2]. Then, instead of defining the elements \overline{x_{1}}, …, \overline{x_{n 1}} , we could have considered a relation T defined on Y_{1} Y_{n 2} U W by

φ_{1}, …, φ_{n 2} , a_{1}, b_{1} T ψ_{1}, …, ψ_{n 2} , a_{2}, b_{2}

if and only if there exists χ_{i} X_{i}, for each 1 i n 1, such that

\begin{matrix}g_{j} χ_{j 1} φ_{j}, & γ χ_{1} a_{1}, \\ f_{j} χ_{j} ψ_{j}, & δ χ_{n 1} b_{2}, \end{matrix}

for all 1 j n 2.

For

c f_{1} x_{1} , f_{2}t_{1}f_{1} x_{1} , …, f_{n 2} t_{n 3} f_{n 3} t_{1}f_{1} x_{1} , γ x_{1} , δt_{n 2} g_{n 2} x_{n 1}

and

d g_{1}s_{2}g_{2} s_{n 2} g_{n 2} x_{n 1} , …, g_{n 3} s_{n 2} g_{n 2} x_{n 1} , g_{n 2} x_{n 1} , γs_{1}f_{1} x_{1} , δ x_{n 1}

we then have:

cT c, by setting χ_{1} x_{1} and χ_{j} t_{j 1} f_{j 1} t_{1}f_{1} x_{1} , 2 j n 1.

Note that δ χ_{n 1} ^{(13)} δt_{n 2} g_{n 2} x_{n 1} .

dT c, by setting χ_{1} s_{1}f_{1} x_{1} and χ_{j} t_{j 1} g_{j 1} s_{j}g_{j} s_{n 2} g_{n 2} x_{n 1} ,

2 j n 1 (here χ_{n 1} t_{n 2} g_{n 2} x_{n 1} ). We have

f_{j} χ_{j} ^{(14)} f_{j}t_{j 1} f_{j 1} t_{1}f_{1} x_{1} for each 2 j n 2.

dT d, by setting χ_{j} s_{j}g_{j} s_{n 2} g_{n 2} x_{n 1} , 1 j n 2, and

χ_{n 1} x_{n 1} . Note that γ χ_{1} ^{(12)} γs_{1}f_{1} x_{1} .

From

c 1_{Y_{1} Y_{n 2} U W}  T T^{}1_{Y_{1} Y_{n 2} U W}  T d

and Theorem 1.3, we obtain cT^{n 1} d, i.e.,

cT ξ_{11}, …, ξ_{1,n 2} , a_{1}, b_{1} T ξ_{21}, …, ξ_{2,n 2} , a_{2}, b_{2}
ξ_{n 3,1} , …, ξ_{n 3,n 2} , a_{n 3} , b_{n 3} T ξ_{n 2,1} , …, ξ_{n 2,n 2} , a_{n 2} , b_{n 2} T d,

for some ξ_{i,j} Y_{i}, 1 i, j n 2, a_{1}, …, a_{n 2} U and b_{1}, …, b_{n 2} W. We then get χ_{11} X_{1}, …, χ_{1,n 1} X_{n 1} such that

\begin{matrix} \\ f_{1} χ_{11} ξ_{11}, …, γ χ_{11} a γ x_{1} , \end{matrix}

χ_{21} X_{1}, χ_{22} X_{2}, …, χ_{2,n 1} X_{n 1} such that

\begin{matrix}g_{1} χ_{22} ξ_{11}, … \\ …, f_{2} χ_{22} ξ_{22}, … \end{matrix}

χ_{n 2,1} X_{1}, …, χ_{n 2,n 2} X_{n 2} , χ_{n 2,n 1} X_{n 1} such that

\begin{matrix}…, g_{n 3} χ_{n 2,n 2} ξ_{n 3,n 3} , … \\ …, f_{n 2} χ_{n 2,n 2} ξ_{n 2,n 2} , … \end{matrix}

χ_{n 1,1} X_{1}, …, χ_{n 1,n 1} X_{n 1} such that

\begin{matrix}…, g_{n 2} χ_{n 1,n 1} ξ_{n 2,n 2} , δ χ_{n 1,n 1} b δ x_{n 1} . \\  \end{matrix}

It now easily follows that χ_{11}, …, χ_{n 1,n 1} L is such that

λ χ_{11}, …, χ_{n 1,n 1} a, b ,

thus λ is a regular epimorphism.

3. Avoiding coproducts

The statement of Theorem 2.3 does not use the hypothesis on the existence of coproducts, while its proof does. A natural question is then the following: Is this assumption really needed? To give a negative answer to this question, we will use the following result.

3.1. Theorem. [17, 14] Let P be an unconditional exactness property. If a small finitely complete category C satisfies P, then Lex C, Set ^{op} also satisfies P.

Here, Lex C, Set denotes the complete and cocomplete category of finite limit preserving functors from C to Set. We recall that the Yoneda embedding C Lex C, Set ^{op} is full and faithful and preserves finite limits and colimits. Roughly speaking, among unconditional exactness properties are properties of the following form: for any diagram whose shape is a fixed finite category, if we build some specified finite (co)limits, then some specified morphism is an isomorphism. For more details, see [17, 14]. Regularity can be expressed as a conjunction of some properties of that form. Let us prove that the same holds for the property

regular 2.3.(iii).

Firstly, let us remove the assumption ‘if γ and δ are regular epimorphisms’ in 2.3.(iii) (since the property has to be ‘unconditional’). This is easy since in a regular context, 2.3.(iii) is equivalent to the following:

(iii’) For any diagram (7) in C, if mp and nq are the (regular epimorphism, monomorphism) factorisations of γ and δ respectively, then the factorisation λ^{}of pπ_{1} and qπ_{n 1} through the pullback of hm and kn is a regular epimorphism.

[figure: L; π_{1}; π_{2}; π_{3}; π_{n 2}; π_{n 1}; λ^{}; I _{V} J; U _{V} W; X; p; f; I; m; hm; U; h; X; f; g; Y; s; t; β; X; g; ; Y; s; t; β; ; X; f; X; g; q; J; n; kn; W; k; Y; s; t; β; ⋮; V; u; w] (16)

Therefore, it remains to prove that the shape of the diagram we started with to build (7) is a finite category. In order to do it, we consider the adjunction

[figure: Cat; U; CondGraph; Path; ]

where CondGraph is the category of small conditional graphs (see Section 5.1 in [3]). If G is a conditional graph and X ob G, we denote by G X the conditional graph constructed from G by removing X and all arrows from or to X. The conditions of G X are all pairs of parallel paths giving rise to the same arrow in Path G . Moreover, if X_{1}, …, X_{n} are objects of G, G X_{1}, …, X_{n} is recursively defined as G X_{1}, …, X_{n 1} X_{n}, with G X_{1} G X_{1}.

3.2. Lemma. Let G be a finite conditional graph and X one of its objects. Suppose that for any pair of arrows f: Y X and g: X Z in G, there exists a path h_{1}, …, h_{n} : Y Z in G X such that the composites gf and h_{n} h_{1} are equal in Path G . Then Path G is finite if and only if Path G X is finite.

Proof. Firstly, we can suppose without loss of generality that all conditions of G X are in G. Indeed, we can add them to the ones of G without changing Path G .

The inclusion morphism G X G gives rise to a faithful functor

Path G X Path G

which shows the ‘only if’ part.

Let us now prove the ‘if part’. If Path G X is finite, there exists an N ℕ such that any morphism of Path G X can be represented by a path of at most N arrows of G X.

Consider a path f_{1}, …, f_{n} in G. By the assumption on X, we know that this path is equal in Path G to a path h_{1}, …, h_{m} where h_{2}, …, h_{m 1} are in G X. Thus we can suppose m 2 N which proves that Path G is finite.

This lemma gives us an easy way to prove that some category generated by a finite conditional graph is finite. We are going to apply it here with the following conditional graph G

\xymatrix{ & X \ar@{=>}[r]^{f} \ar@{=>}[r]|{g} \ar@{=>}[d]_{δ} \ar@{=>}[ddl]_{γ} & Y \ar@{=>}[l]|{s} \ar@{=>}[l]^{t} \ar@{=>}[d]_{β} \\  & W \ar@{=>}[r]^{k} & V \ar@{=>}[l]^{w} \ar@{=>}[u]_{i} \ar@{=>}[u]_{i^{}} \ar@{=>}[dll]^{u} \\ U \ar@{=>}[urr]|{h}}

where the conditions are

fs gt 1_{Y} , βi βi^{}kw hu 1_{V} , βg βf hγ kδ,
γs uβ, δt wβ, ft iβ and gs i^{}β.

Notice that diagram (16) is build up from Path G by adding some finite (co)limits to it. Due to the equalities hγ βf and hu 1_{V}, Path G is finite if and only if Path G U is. Since kδ βg and kw 1_{V}, it suffices to consider the category Path G U, W . Then, with ft iβ and gs i^{}β, we only have to prove that Path G U, W, V is finite. Since fs gt 1_{Y}, if we add some formal arrows y, y^{}: Y Y and the conditions ft y and gs y^{}, we only have to prove that the category generated by

[figure: Y; y; y^{}]

and the equalities coming from G is finite. But this is obvious since yy ftft iβiβ iβ y, yy^{}y, y^{}y y^{}and y^{}y^{}y^{}.

We can thus remove the hypothesis about coproducts in Theorem 2.3.

3.3. Theorem. Let n 3 and let C be a regular category. The following statements are equivalent:

(i) C is an n-permutable category;

(ii) if γ and δ in diagram (5) are regular epimorphisms, then the comparison morphism λ is also a regular epimorphism;

(iii) if γ and δ in diagram (7) are regular epimorphisms, then the comparison morphism λ is also a regular epimorphism.

Proof. (i) (ii) (iii) has been proved for Theorem 2.3. Let us prove (iii) (i). Up to a change of universe, we can suppose C to be small. Since

regular (iii) regular (iii’)

is the conjunction of some unconditional exactness properties, we know from Theorem 3.1 that Lex C, Set ^{op} is also a regular category which satisfies (iii). Since it has small colimits, we deduce from Theorem 2.3 that it is n-permutable. Since the embedding C Lex C, Set ^{op} is full, faithful and preserves finite limits and colimits, C is also npermutable.

4. The ternary terms for n-permutable varieties

The stability property for regular epimorphisms given by Theorem 3.3 for n-permutable varieties gives us a diagram that exhibits the existence of the ternary operations which characterise them. In fact, let X be the free algebra on one element and consider diagram (7) with f δ ∇_{X} 1_{X}, s ι_{2} ι_{3} , g γ 1_{X} ∇_{X}, t ι_{1} ι_{2} , β k h ∇_{X}, w i ι_{1} and u i^{}ι_{2}. The binary terms p_{1} x, y x and p_{2} x, y y are such that p_{1}, p_{2} Eq ∇_{X} . Since λ is surjective, then there exist ternary operations w_{1}, …, w_{n 1} L such that λ w_{1}, …, w_{n 1} p_{1}, p_{2} . The identities from Theorem 1.1 then easily follow:

- since w_{1}, …, w_{n 1} belongs to the limit L, we obtain the identities w_{1} x, x, y w_{2} x, y, y , w_{2} x, x, y w_{3} x, y, y , …, w_{n 2} x, x, y w_{n 1} x, y, y ;

- from λ w_{1}, …, w_{n 1} p_{1}, p_{2} , we get 1_{X} ∇_{X} w_{1} p_{1} and ∇_{X} 1_{X} w_{n 1} p_{2} and, consequently, w_{1} x, y, y x and w_{n 1} x, x, y y.

5. Another stability property and the n 1)-ary operations

We saw how the stability properties for regular epimorphisms stated in Theorem 3.3 provided a diagram which exhibits the existence of ternary operations for the varietal case. The following question arises naturally:

Is there a (stability) property for n-permutable categories which exhibits the existence of the n 1 -ary operations for the varietal case?

We conclude this paper with a positive answer; the proof is done separately for n 3 congruent to 0, 1, 2 and 3, modulo 4.

For n 3 odd, we consider the diagram

[figure: L; π_{1}; π_{2}; π_{3}; π_{n 2}; π_{n 1}; λ; Eq h; q_{2}; q_{1}; X; g; f; Z; h; X; g; f; Y; s; s; X; g; ; Z; t; t; ; X; f; X; f; g; Z; h; Y; s; s; k; V] (17)

where the left side is a limit of split epimorphisms and the right quadrangle is the kernel pair of h. We also assume kf hg, which implies the existence of a comparison morphism λ: L Eq h such that q_{1}λ gπ_{1} and q_{2}λ gπ_{n 1} . The morphisms f, g, h and k are supposed to be regular epimorphisms.

5.1. Theorem. Let n 4l 3 3 be congruent to 3 modulo 4 and let C be a regular category. The following statements are equivalent:

(i) C is an n-permutable category;

(ii) for any diagram (17), where h is split by a morphism u such that

uk gsft ^{l}gs gs ftgs ^{l},  (18)

then the comparison morphism λ is a regular epimorphism.

Proof. Note that if n 3, then the equivalence between (i) and (ii) is precisely Theorem 1 in [10]. Suppose now that n 4l 3 7.

(i) (ii) As in Theorem 2.3, we are going to use the embedding theorem for npermutable categories, this time using the category Mod Γ_{n}^{}. So let s^{}S_{n}^{}. If a, b Eq h _{s^{}}, then we can suppose there exist x_{1} X_{s^{}}and x_{n 1} X_{s^{}}such that g x_{1} a and g x_{n 1} b. Since h a h b , we get

kf x_{1} hg x_{1} hg x_{n 1} kf x_{n 1} .  (19)

We consider the elements

\overline{x_{i}} v_{i}^{s^{}}x_{1}, sf x_{1} , tgsf x_{1} , …, sf tgsf ^{l} x_{1} , sf tgsf ^{l} x_{n 1} , …, sf x_{n 1} , x_{n 1} in X_{s^{},0}, for each 1 i n 1. Note that the first half 2l 2 elements (an even number) is calculated using x_{1} and the second half 2l 2 elements is calculated using x_{n 1} . We want to show that \overline{x_{1}}, …, \overline{x_{n 1}} L_{s^{},0} is such that λ \overline{x_{1}}, …, \overline{x_{n 1}} α^{s^{}}a, b . Note that, for any 1 i n 1,

f \overline{x_{i}} v_{i}^{s^{}}f x_{1} , f x_{1} , ftgsf x_{1} , ftgsf x_{1} ,…, f tgsf ^{l} x_{1} , f tgsf ^{l} x_{1} ,
f tgsf ^{l} x_{n 1} , f tgsf ^{l} x_{n 1} , …, f x_{n 1} , f x_{n 1} ,
g \overline{x_{i}} v_{i}^{s^{}}g x_{1} , gsf x_{1} , gsf x_{1} , …, gsf tgsf ^{l} x_{1} ,
gsf tgsf ^{l} x_{n 1} , …, gsf x_{n 1} , gsf x_{n 1} , g x_{n 1} ,

and

gsf tgsf ^{l} x_{1} gsft ^{l}gsf x_{1} ^{(18)} ukf x_{1} ^{(19)} ukf x_{n 1} gsf tgsf ^{l} x_{n 1} . (20)

From the identities (2), we obviously have f \overline{x_{i 1}} f \overline{x_{i}} , when i is even, and to obtain g \overline{x_{i 1}} g \overline{x_{i}} , when i is odd, we use (20). We also need (20) to get g \overline{x_{1}} α^{s^{}}g x_{1} α^{s^{}}a and g \overline{x_{n 1}} α^{s^{}}g x_{n 1} α^{s^{}}b . Therefore,

a, b ^{2} π^{s^{}}α^{s^{}}a, b π^{s^{}}λ \overline{x_{1}}, …, \overline{x_{n 1}} Im λ _{s}

and λ is a regular epimorphism.

(ii) (i) The proof is similar to the one of (iii) (i) in Theorem 3.3. First we prove it using binary coproducts and Theorem 5 of [24] (characterising n-permutable categories with binary coproducts through n 1 -ary approximate Hagemann-Mitschke co-operations). For an arbitrary object X, we apply the assumption to diagram (17) with

f 2l 2 ∇_{X}: 4l 4 X 2l 2 X,
s ι_{2} ι_{4} ι_{2l} ι_{2l 2} ι_{2l 3} ι_{2l 5} ι_{4l 3} ,
g 1_{X} 2l 1 ∇_{X} 1_{X}: 4l 4 X 2l 3 X,
t ι_{1} ι_{3}  ι_{2l 1} ι_{2l 3} ι_{2l 4} ι_{2l 6}  ι_{4l 4} ,
k ∇_{X}^{2l 2} : 2l 2 X X,
h ∇_{X}^{2l 3} : 2l 3 X X

and u ι_{l 2} , where ∇_{X}^{i}: iX X denotes the codiagonal.

We must check that uk gsft ^{l}gs. We are actually going to check it in the dual category C^{op} using a set-theoretical argument. For that, we consider x_{1}, …, x_{2l 3} in the product X^{2l 3} and the effect of applying the first gsft:

x_{1}, …, x_{2l 3} ^{gsft} x_{2}, …, x_{l 1} , x_{l 2} , x_{l 2} , x_{l 2} , x_{l 3} , …, x_{2l 2} .

The first and last entries disappear while the middle one appears three times. Thus

x_{1}, …, x_{2l 3} ^{gsft ^{l}} x_{l 1} , x_{l 2} , …, x_{l 2} , x_{l 3} ^{gs} x_{l 2} , …, x_{l 2} .

The rest of the proof using binary coproducts then goes similarly as in Theorem 2.3.

To prove that we can remove this assumption on coproducts, we use the same technique as in Section 3. The only non-obvious part is to prove that the category underlying the shape of the diagram used to build diagram (17) together with the section u of h as in (ii) is finite. The corresponding conditional graph G is given by

\xymatrix{X \ar@{=>}[r]_{g} \ar@{=>}[d]_{f} & Z \ar@{=>}[l]_{t} \ar@{=>}[d]_{h} \\ Y \ar@{=>}[u]_{s} \ar@{=>}[r]_{k} & V \ar@{=>}[u]_{u}}

with the conditions fs 1_{Y}, gt 1_{Z}, hu 1_{V}, hg kf and uk gsft ^{l}gs. Obviously, we can add some arrows y: Y Z and z: Z Y together with the conditions y gs and z ft to form a conditional graph G^{}such that Path G Path G^{}. Since fs 1_{Y}, gs y, ft z and gt 1_{Z}, applying Lemma 3.2, Path G^{}is finite if and only if Path G^{}X is. Again, we can add an arrow x: Z Z and the condition x yz to G^{}X to obtain the conditional graph G^{}such that Path G^{}Path G^{}X .

[figure: Z; h; x; z; Y; k; y; V; u]

Since kz kft hgt h and yz x, we only need to show that Path G^{}Y is finite. Again, since uh ukft gsft ^{l}gsft gsft ^{l 1} yz ^{l 1} x^{l 1} , it remains to show that Path G^{}Y, V is finite. For that, it is enough to observe that

x^{2l 2} gsft ^{2l 2} gsft ^{l}gsft gsft ^{l 1}
ukft gsft ^{l 1} uh gsft ^{l}gsft
uhukft ukft gsft ^{l 1} x^{l 1} .

5.2. Remark. If V is an n-permutable variety of universal algebras (n 4l 3), the above stability property exhibits the n 1 -ary operations from Theorem 1.2. Indeed, let X be the free algebra on one element and consider diagram (17) together with the splitting u of h as given in the proof of (ii) (i). The 2l 3 -ary terms p_{1} and p_{2l 3} given by the first and the 2l 3 -th projections, respectively, are such that p_{1}, p_{2l 3} Eq ∇_{X}^{2l 3} . Since λ is a surjection, by assumption, there exist 4l 4 n 1 -ary terms v_{1}, …, v_{n 1} L such that λ v_{1}, …, v_{n 1} p_{1}, p_{2l 3} . It then easily follows that v_{1}, …, v_{n 1} satisfy the identities of Theorem 1.2.

5.3. Theorem. Let n 4l 1 5 be congruent to 1 modulo 4 and let C be a regular category. The following statements are equivalent:

(i) C is an n-permutable category;

(ii) for any diagram (17) where k is split by a morphism w such that

wk ftgs ^{l},  (21)

the comparison morphism λ is a regular epimorphism.

Proof. The proof is similar to that of Theorem 5.1, where now

\overline{x_{i}} v_{i}^{s^{}}x_{1}, sf x_{1} , tgsf x_{1} , …, tgsf ^{l} x_{1} , tgsf ^{l} x_{n 1} , …, sf x_{n 1} , x_{n 1}

is used for the implication (i) (ii). For the converse implication (ii) (i), we consider the following morphisms

f 2l 1 ∇_{X}: 4l 2 X 2l 1 X,
s ι_{2} ι_{4} ι_{2l} ι_{2l 2} ι_{2l 3} ι_{2l 5} ι_{4l 1} ,
g 1_{X} 2l ∇_{X} 1_{X}: 4l 2 X 2l 2 X,
t ι_{1} ι_{3}  ι_{2l 1} ι_{2l 1} ι_{2l 2} ι_{2l 4}  ι_{4l 2} ,
k ∇_{X}^{2l 1} : 2l 1 X X,
h ∇_{X}^{2l 2} : 2l 2 X X

and w ι_{l 1} .

For n 4 even, we consider the diagram

[figure: L; π_{1}; π_{2}; π_{3}; π_{n 2}; π_{n 1}; λ; Z _{V} Y; q_{2}; q_{1}; X; g; f; Z; h; X; g; f; Y; s; s; X; g; ; Z; t; t; ; X; g; X; g; f; Y; k; Z; t; t; h; V] (22)

where the left side is a limit of split epimorphisms and the right quadrangle is a pullback of regular epimorphisms. We also assume kf hg, which implies the existence of a comparison morphism λ: L Z _{V} Y such that q_{1}λ gπ_{1} and q_{2}λ fπ_{n 1} .

5.4. Theorem. Let n 4l 4 be divisible by 4 and let C be a regular category. The following statements are equivalent:

(i) C is an n-permutable category;

(ii) for any diagram (22), where k is split by a morphism w such that

wk ftgs ^{l},  (23)

then the comparison morphism λ is a regular epimorphism.

Proof. The proof is similar to that of Theorem 5.1, where now

\overline{x_{i}} v_{i}^{s^{}}x_{1}, sf x_{1} , tgsf x_{1} , …, tgsf ^{l} x_{1} , tgsf ^{l} x_{n 1} , …, sf x_{n 1}

is used for the implication (i) (ii). For the converse implication (ii) (i), we consider the following morphisms

f 2l ∇_{X} 1_{X}: 4l 1 X 2l 1 X,
s ι_{2} ι_{4} ι_{2l 2} ι_{2l} ι_{2l 1} ι_{2l 3} ι_{4l 1} ,
g 1_{X} 2l ∇_{X}: 4l 1 X 2l 1 X,
t ι_{1} ι_{3}  ι_{2l 1} ι_{2l 1} ι_{2l 2} ι_{2l 4}  ι_{4l} ,
k ∇_{X}^{2l 1} : 2l 1 X X,
h ∇_{X}^{2l 1} : 2l 1 X X

and w ι_{l 1} .

5.5. Theorem. Let n 4l 2 6 be congruent to 2 modulo 4 and let C be a regular category. The following statements are equivalent:

(i) C is an n-permutable category;

(ii) for any diagram (22), where h is split by a morphism u such that

uk gsft ^{l}gs gs ftgs ^{l},  (24)

then the comparison morphism λ is a regular epimorphism.

Proof. The proof is similar to that of Theorem 5.1, where now

\overline{x_{i}} v_{i}^{s^{}}x_{1}, sf x_{1} , tgsf x_{1} , …, sf tgsf ^{l} x_{1} , sf tgsf ^{l} x_{n 1} , …, sf x_{n 1}

is used for the implication (i) (ii). For the converse implication (ii) (i), we consider the following morphisms

f 2l 1 ∇_{X} 1_{X}: 4l 3 X 2l 2 X,
s ι_{2} ι_{4} ι_{2l} ι_{2l 2} ι_{2l 3} ι_{2l 5} ι_{4l 3} ,
g 1_{X} 2l 1 ∇_{X}: 4l 3 X 2l 2 X,
t ι_{1} ι_{3}  ι_{2l 1} ι_{2l 1} ι_{2l 2} ι_{2l 4}  ι_{4l 2} ,
k ∇_{X}^{2l 2} : 2l 2 X X,
h ∇_{X}^{2l 2} : 2l 2 X X

and u ι_{l 2} .

5.6. Remark. As for the case n 4l 3, the above characterisations of n-permutable categories for the cases n 4l, 4l 1 and 4l 2 also give rise to the n 1 -ary operations in the case of an n-permutable variety of universal algebras.

References

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[24] D. Rodelo, T. Van der Linden, Approximate Hagemann–Mitschke co-operations, Appl. Categ. Structures 22 (2014) 1009-1020.

Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, chemin du cyclotron 2 bte L7.01.02, B–1348 Louvain-la-Neuve, Belgium

CMUC, Department of Mathematics, University of Coimbra, 3001–501 Coimbra, Portugal and Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005–139 Faro, Portugal

Email: pierre-alain.jacqmin@uclouvain.be
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