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Let (hyp o n , ♯) be the hypoplactic monoid of finite rank n with Schützenberger's involution ♯ . In this paper, we exhibit a faithful representation of (hyp o n , ♯) 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 (hyp o n , ♯) . Further, we prove that (hyp o n , ♯) 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 (hyp o n , ♯) . Communicated by Scott Chapman [ABSTRACT FROM AUTHOR]

Let (baxt n , ♯) be the Baxter monoid of finite rank n with Schützenberger's involution ♯ . In this paper, it is shown that (baxt n , ♯) admits a faithful representation by an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. Then a transparent combinatorial characterization of the word identities satisfied by (baxt n , ♯) is given. Further, it is proved that (baxt n , ♯) is finitely based if and only if n ≠ 3 , and shown that the identity checking problem for (baxt n , ♯) is decidable in polynomial time. [ABSTRACT FROM AUTHOR]