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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]

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]

Abstract 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 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]