A Note on Semigroup Algebras of Permutable Semigroups
Attila Nagy, M\'arton Zubor

TL;DR
This paper investigates the conditions under which a specific mapping from ideals of a semigroup algebra to congruences forms a homomorphism, establishing a link with permutability of the underlying semigroup.
Contribution
It characterizes permutability of semigroups via the homomorphic property of a map related to their semigroup algebras, focusing on semilattices and rectangular bands.
Findings
The map is a homomorphism if and only if the semigroup is permutable.
The result applies specifically to semilattices and rectangular bands.
Provides a new algebraic characterization of permutability in semigroup theory.
Abstract
Let be a semigroup and be a field. For an ideal of the semigroup algebra of over , let denote the restriction (to ) of the congruence on defined by the ideal . A semigroup is called a permutable semigroup if is satisfied for all congruences and of . In this paper we show that if is a semilattice or a rectangular band then is a homomorphism of the semigroup into the relations semigroup if and only if is a permutable semigroup.
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Taxonomy
Topicssemigroups and automata theory · Advanced Algebra and Logic · Rings, Modules, and Algebras
