ﻻ يوجد ملخص باللغة العربية
We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of $ntimes n$ upper triangular tropical matrices, in terms of equivalence of certain tropical polynomials. This leads to an algorithm for checking whether such an identity holds, in time polynomial in the length of the identity and size of the alphabet. It also allows us to answer a question of Izhakian and Margolis, by showing that the identities which hold in the monoid of $2times 2$ upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid. Our results extend to a broader class of chain structured tropical matrix semigroups; we exhibit a faithful representation of the free monogenic inverse semigroup within such a semigroup, which leads also to a representation by $3times 3$ upper triangular matrix semigroups, and a new proof of the fact that this semigroup satisfies the same identities as the bicyclic monoid.
This paper presents new results on the identities satisfied by the hypoplactic monoid. We show how to embed the hypoplactic monoid of any rank strictly greater than 2 (including infinite rank) into a direct product of copies of the hypoplactic monoid
We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A conseq
The multiplicative semigroup $M_n(F)$ of $ntimes n$ matrices over a field $F$ is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form $M_n(S)$, where $S$ is a semiring, and the subsemigroups $UT_n(S)$
The 6-element Brandt monoid $B_2^1$ admits a unique addition under which it becomes an additively idempotent semiring. We show that this addition is a term operation of $B_2^1$ as an inverse semigroup. As a consequence, we exhibit an easy proof that
We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and Develin i