Representations and identities of hypoplactic monoids with involution

AbstractLet (hypon,♯) be the hypoplactic monoid of finite rank n with Schützenberger’s involution ♯. In this paper, we exhibit a faithful representation of (hypon,♯) as an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. We then give a transparent combinatorial characterization of the word identities satisfied by (hypon,♯). Further, we prove that (hypon,♯) is non-finitely based if and only if n = 2, 3 and give a polynomial time algorithm to check whether a given word identity holds in (hypon,♯).Communicated by Scott ChapmanKEYWORDS: Finite basis problemidentityhypoplactic monoidrepresentationSchützenberger’s involution2020 MATHEMATICS SUBJECT CLASSIFICATION: 20M0720M3005E9912K1016Y60 AcknowledgmentsThe authors are very grateful to the anonymous referee for his/her careful reading and suggestions.Additional informationFundingThis research was partially supported by the National Natural Science Foundation of China (Nos. 12271224, 12171213, 12161062), the Fundamental Research Funds for the Central University (No. lzujbky-2023-ey06) and the Natural Science Foundation of Gansu Province (No. 23JRRA1055).