Let M be a set. A set-theoretical solution of the pentagon equation on M is a map s : M × M → M × M such that s23 s13 s12 = s12 s23 where (Formula presented.) and (Formula presented.) and τ is the flip map, i.e., the permutation on M × M given by (Formula presented.) for all x,y ϵ M. In this paper, we give a complete description of the set-theoretical solutions of the form (x.y,x*y) when either (m, .) or (M, *) is a group; moreover, we raise some questions.
Set-theoretical solutions of the pentagon equation on groups
Catino F.;Mazzotta M.;Miccoli M. M.
2020-01-01
Abstract
Let M be a set. A set-theoretical solution of the pentagon equation on M is a map s : M × M → M × M such that s23 s13 s12 = s12 s23 where (Formula presented.) and (Formula presented.) and τ is the flip map, i.e., the permutation on M × M given by (Formula presented.) for all x,y ϵ M. In this paper, we give a complete description of the set-theoretical solutions of the form (x.y,x*y) when either (m, .) or (M, *) is a group; moreover, we raise some questions.File in questo prodotto:
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