Associative pentagon algebras

Abstract A set-theoretic solution to the Pentagon Equation can be described as a pentagon algebra $$(S, \cdot , *)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mo>·</mml:mo> <mml:mo>,</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> such that $$(S, \cdot )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a semigroup and the operations $$\cdot $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>·</mml:mo> </mml:math> and $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> are related by two additional equations. This paper aims to investigate associative pentagon algebras in which $$(S, *)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is also a semigroup. We introduce and describe two families of associative pentagon algebras which are strongly determined by the properties of the semigroup $$(S,*)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . 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.