The pentagon equation classically originates from the field of Mathematical Physics. Our attention is placed on the study of set-theoretical solutions of this equation, namely, maps s: X×X → X×X given by s(x, y)=(xy, θx(y)), where X is a semigroup and θx: X → X is a map satisfying two laws. In this talk, we give some recent descriptions of some classes of solutions achieved starting from particular semigroups. Into the specific, we provide a characterization of idempotent-invariant solutions on a Clifford semigroup X, that are those for which θa remains invariant on the set of idempotents E(X). In addition, we will focus on the classes of involutive and idempotent solutions, which are solutions fulfilling s2=idX×X and s2=s, respectively.
This talk relates to this arXiv paper.
This video is part of the European Non-Associative Algebra Seminar series.
