## Finite basis problem for Catalan monoids with involution.(English)Zbl 1509.20071

Summary: It is known, since the early 2000s, that the Catalan monoid $$C_n$$ generated by $$n$$ elements is finitely based if and only if $$n\leq 3$$. The main goal of this paper is to prove that the involution monoid $$( C_3,{}^*)$$ is non-finitely based. Therefore, in contrast, combining with previous results yields that the involution Catalan monoid $$( C_n,{}^*)$$ is finitely based if and only if $$n=1$$.
The Kiselman monoid $$K_n$$ generated by $$n$$ elements is also considered. Although the semigroups $$C_n$$ and $$K_n$$ have recently been shown to satisfy the same identities, it is unknown if the same result holds when they are considered as involution semigroups. Nevertheless, it is deduced from the main result that the involution Kiselman monoid $$( K_n,{}^*)$$ is finitely based if and only if $$n=1$$.

### MSC:

 20M05 Free semigroups, generators and relations, word problems 20M20 Semigroups of transformations, relations, partitions, etc.

### Keywords:

semigroup; monoid; involution; Catalan monoid; finite basis problem
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### References:

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