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Let (프n,ρ) be the involution monoid of Annular monoid 프n under the rotation involution ρ. The involution monoids (프1,ρ) and (프2,ρ) are easily seen to be finitely based; Auinger et al. proved that (프n,ρ) is inherently non-finitely based if n ≥ 4. In this paper, we show that (프3,ρ) is finitely based by providing a finite identity basis for (프3,ρ), which answers an open question posed by Auinger et al. Therefore, the involution monoid (프n,ρ) is finitely based if and only if n ≤ 3. [ABSTRACT FROM AUTHOR]

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]

In this paper, we show that all stalactic and taiga monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities, that all sylvester monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities and that all baxter monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities. [ABSTRACT FROM AUTHOR]