On the construction of Noetherian forms for algebraic structures

Francois Koch van Niekerk

A Noetherian form is a self-dual axiomatic context in which the Noether isomorphism theorems and other homomorphism theorems can be established. These theorems for group-like algebraic structures (for example groups, rings without unity and vector spaces) can be obtained by choosing a Noetherian form based on lattices of subalgebras. In this paper we show that by replacing lattices of subalgebras with some other lattices, it becomes possible to move beyond group-like structures and encompass all types of algebraic structures (including sets, monoids, lattices). Moreover, we show that in a suitable sense, existence of a Noetherian form for a give type of mathematical structure is intimately linked with algebraicity of structures. The isomorphism theorems resulting from applying these Noetherian forms recover the isomorphism theorems known for general algebraic structures in the literature.

Keywords: Isomorphism Theorems, Monads, Noetherian Forms, Varieties

2020 MSC: 08C05, 08A30, 18C15, 18D99

Theory and Applications of Categories, Vol. 39, 2023, No. 8, pp 189-206.

Published 2023-03-01.


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