Abstract
It is shown that the plactic monoid M of rank $$3$$ satisfies the identity $$wvvwvw=wvwvvw$$ where $$v=xyyx xy xyyx$$ and $$w= xyyx yx xyyx$$ . This is accomplished by first showing that certain simple monoids related to $$M$$ satisfy this identity. These simple monoids are natural generalizations of the bicyclic monoid $$B$$ , which satisfies the identity $$w=v$$ by a result of Adjan.
Highlights
For an integer n ≥ 1 we consider the finitely presented monoid Mn = a1, . . . , an defined by the relations ai ak a j = ak ai a j for i ≤ j < k, a j ai ak = a j ak ai for i < j ≤ k
A motivation comes from general problems concerning existence of identities in classes of finitely generated semigroups of polynomial growth
While simple monoids seem to be of special interest from the point of view of varieties of semigroups, it is worth mentioning that the structure of Mn heavily depends on certain simple monoids discovered in [10]
Summary
Conjecture The plactic monoid of any rank n ≥ 1 satisfies a nontrivial identity. We start with a simple conceptual proof of this result, and for a more detailed study of identities satisfied by the bicyclic monoid we refer to [16]. Proposition 2.1 The bicyclic monoid B satisfies the identity w(x, y) = v(x, y).
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