[Paper Review] On disjunctions of equations over finite simple semigroups

Аrtеm N. Shevlyakov|arXiv (Cornell University)|May 24, 2013

semigroups and automata theory1 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for a finite simple semigroup to be an equational domain—meaning finite unions of algebraic sets are algebraic—using its Rees matrix representation. It proves that finite simple semigroups with trivial center can be equational domains, while any semigroup with a nontrivial center (including those with zero, commutative ones, or non-group homogroups) cannot be an equational domain, resolving a key question in algebraic geometry over semigroups.

ABSTRACT

A semigroup $S$ is called an equational domain if any finite union of algebraic sets over $S$ is algebraic. For a finite simple semigroup we find necessary and sufficient conditions to be an equational domain. Moreover, we study semigroups with nontrivial center and prove that any such semigroup is not an equational domain.

Motivation & Objective

• To determine necessary and sufficient conditions for a finite simple semigroup to be an equational domain.
• To resolve whether there exists a nontrivial semigroup that is an equational domain but not a group.
• To extend known results on equational domains from groups to semigroups, particularly focusing on finite simple semigroups.
• To investigate the role of the center in semigroups regarding the equational domain property.
• To establish that adjunction of an identity to a finite simple equational domain preserves the equational domain property.

Proposed method

• Represent finite simple semigroups via Rees matrix semigroups $S = (G, \mathbf{P}, \Lambda, I)$, where $G$ is a finite group and $\mathbf{P}$ is a normalized matrix over $G$.
• Use the structure of algebraic sets defined by systems of equations $t(X) = s(X)$ over semigroups, where terms are products of variables and semigroup elements.
• Apply the concept of equational domains as semigroups where finite unions of algebraic sets are algebraic.
• Construct counterexamples using ideals and central elements to show non-algebraicity of certain disjunctions.
• Use commutativity arguments and element-level computations to derive contradictions when assuming certain disjunctions are algebraic.
• Leverage the fact that in finite groups, negative exponents can be replaced by positive ones via $x^{-1} = x^{|G|-1}$, enabling translation of group equations into semigroup equations.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a finite simple semigroup to be an equational domain?
• RQ2 Can a nontrivial semigroup that is not a group be an equational domain?
• RQ3 How does the presence of a nontrivial center affect the equational domain property in semigroups?
• RQ4 Does the adjunction of an identity element preserve the equational domain property in finite simple semigroups?
• RQ5 Are there semigroups with zero or commutative semigroups that can be equational domains?

Key findings

• A finite simple semigroup $S = (G, \mathbf{P}, \Lambda, I)$ is an equational domain if and only if $|\Lambda| = |I| = 1$ or $G$ is trivial and $|\Lambda| = |I| = 1$, i.e., $S$ is a group.
• The paper constructs an explicit example of a non-group finite simple semigroup that is an equational domain, confirming a positive solution to the problem of nontrivial non-group equational domains.
• The adjunction of an identity element to a finite simple equational domain semigroup preserves the equational domain property.
• Any semigroup with a nontrivial center is not an equational domain if there exists an element $a$ such that $ae \neq a$ for a central element $e$.
• Nontrivial semigroups with zero, commutative semigroups, and non-group homogroups are not equational domains.
• The kernel of a homogroup is a group, and if the semigroup is an equational domain, then it must be a group (i.e., $S = Ker(S)$).