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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]

An example of a nonfinitely based involution monoid of order five has recently been discovered. We confirm that this example is, up to isomorphism, the unique smallest among all involution monoids. [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]

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]

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]

Let M n be the variety generated by all monoids of order n. It was known that M n is finitely based if n ⩽ 4 and non-finitely based if n ⩾ 6. However, the finite basis problem for M 5 remained open. In this paper, we show that the variety M 5 is finitely based by giving a finite identity basis. Therefore, M n is finitely based if and only if n ⩽ 5. [ABSTRACT FROM AUTHOR]