## Abstract

A set-theoretic solution of the Pentagon Equation on a non-empty set S is a map s: S^{2}→ S^{2} such that s_{23}s_{13}s_{12}= s_{12}s_{23}, where s12=s×id, s23=id×s and s13=(τ×id)(id×s)(τ×id) are mappings from S^{3} to itself and τ: S^{2}→ S^{2} is the flip map, i.e., τ(x, y) = (y, x). We give a description of all involutive solutions, i.e., s2=id. It is shown that such solutions are determined by a factorization of S as direct product X× A× G and a map σ:A→Sym(X), where X is a non-empty set and A, G are elementary abelian 2-groups. Isomorphic solutions are determined by the cardinalities of A, G and X, i.e., the map σ is irrelevant. In particular, if S is finite of cardinality 2 ^{n}(2 m+ 1) for some n, m⩾ 0 then, on S, there are precisely (n+22) non-isomorphic solutions of the Pentagon Equation.

Original language | English |
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Pages (from-to) | 1003-1024 |

Number of pages | 22 |

Journal | Communications in Mathematical Physics |

Volume | 380 |

Issue number | 2 |

DOIs | |

Publication status | Published - Dec 2020 |