A set-theoretic solution to the Pentagon Equation can be described as a pentagon algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(S, \cdot , )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mo>·</mo> <mo>,</mo> <mrow /> <mo>∗</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(S, \cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a semigroup and the operations <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\cdot $</EquationSource> <EquationSource Format="MATHML"><math> <mo>·</mo> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation> are related by two additional equations. This paper aims to investigate associative pentagon algebras in which <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(S, )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mrow /> <mo>∗</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is also a semigroup. We introduce and describe two families of associative pentagon algebras which are strongly determined by the properties of the semigroup <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(S,)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mrow /> <mo>∗</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We present a complete characterization of such algebras using semigroup equations. We also provide constructions of such associative pentagon algebras and give several classes of examples.

Associative pentagon algebras

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2026年6月16日

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Marzia Mazzotta,
Agata Pilitowska

摘要

A set-theoretic solution to the Pentagon Equation can be described as a pentagon algebra $(S, \cdot , *)$ $(S, \cdot, *)$ such that $(S, \cdot )$ $(S, \cdot)$ is a semigroup and the operations $\cdot $ $\cdot$ and $*$ $*$ are related by two additional equations. This paper aims to investigate associative pentagon algebras in which $(S, *)$ $(S, *)$ is also a semigroup. We introduce and describe two families of associative pentagon algebras which are strongly determined by the properties of the semigroup $(S,*)$ $(S, *)$ . We present a complete characterization of such algebras using semigroup equations. We also provide constructions of such associative pentagon algebras and give several classes of examples.