Semigroup Algebra Structures and Their Applications
Summary
Semigroup algebras arise by extending an associative semigroup into a linear framework over a chosen field, thereby uniting combinatorial and algebraic perspectives. Central to their study is the interplay between the underlying semigroup’s idempotent structure and the representation theory of the resulting algebra. Key themes include decomposition theorems, where multiplicative bases built from idempotents yield direct‐sum or direct‐product decompositions of module categories, and conditions under which these algebras become semisimple or self‐injective. Applications span automata theory, where transition semigroups encode state machines; coding and cryptography, via algebraic coding alphabets modelled on monoid actions; and mathematical physics, through operator semigroups in evolution equations. Current research focuses on characterising finiteness and regularity conditions that guarantee desirable homological properties, as well as on extending classical results—such as Green’s relations and Rees matrix constructions—to broader classes of abundant or adequate semigroups. These advances illuminate how local combinatorial data govern global algebraic behaviour, offering practical tools for both pure and applied disciplines.
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Semigroup Algebra Structures and Their Applications publication trend
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Technical terms
Semigroup: A set equipped with an associative binary operation.
Semigroup algebra: The vector space over a field spanned by semigroup elements, endowed with multiplication induced by the semigroup law.
Idempotent: An element e of a semigroup satisfying e·e = e, central to decomposition techniques.
Self-injective algebra: An algebra for which every homomorphism from a submodule to the algebra extends to the entire module.
Semisimple algebra: An algebra that decomposes as a direct sum of simple modules, exhibiting no non-zero nilpotent ideals.
References
- Self-injectivity of semigroup algebras. Open Mathematics (2020).
- Locally adequate semigroup algebras. Open Mathematics (2016).
- On graph products of monoids. Journal of Algebra (2023).
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