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Register a new userSemi-biproducts in monoids
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Authors
Nelson Martins-Ferreira
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It is shown that the category of semi-biproducts in monoids is equivalent to a category of pseudo-actions. A semi-biproduct in monoids is at the same time a generalization of a semi-direct product in groups and a biproduct in commutative monoids. Every Schreier extension of monoids can be seen as an instance of a semi-biproduct; namely a semi-biproduct whose associated pseudo-action has a trivial correction system. A correction system is a new ingredient that must be inserted in order to obtain a pseudo-action out of a pre-action and a factor system. In groups, every correction system is trivial. Hence, semi-biproducts there are the same as semi-direct products with a factor system, which are nothing but group extensions. An attempt to establish a general context in which to define semi-biproducts is made. As a result, a new structure of map-transformations is obtained from a category with a 2-cell structure. Examples and first basic properties are briefly explored.
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It is shown that the category of emph{semi-biproducts} of monoids is equivalent to the category of emph{pseudo-actions}. A semi-biproduct of monoids is a new notion, obtained through generalizing a biproduct of commutative monoids. By dropping commutativity and requiring some of the homomorphisms in the biproduct diagram to be merely identity-preserving maps, we obtain a semi-biproduct. A pseudo-action is a new notion as well. It consists of three ingredients: a pre-action, a factor system and a correction system. In the category of groups all correction systems are trivial. This is perhaps the reason why this notion, to the authors best knowledge, has never been considered before.
We establish a criterion for a semigroup identity to hold in the monoid of $n times n$ upper unitriangular matrices with entries in a commutative semiring $S$. This criterion is combinatorial modulo the arithmetic of the multiplicative identity element of $S$. In the case where $S$ is idempotent, the generated variety is the variety $mathbf{J_{n-1}}$, which by a result of Volkov is generated by any one of: the monoid of unitriangular Boolean matrices, the monoid $R_n$ of all reflexive relations on an $n$ element set, or the Catalan monoid $C_n$. We propose $S$-matrix analogues of these latter two monoids in the case where $S$ is an idempotent semiring whose multiplicative identity element is the `top element with respect to the natural partial order on $S$, and show that each generates $mathbf{J_{n-1}}$. As a consequence we obtain a complete solution to the finite basis problem for lossy gossip monoids.
In 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type $A$). In this paper, we give a family of presentations of Boolean reflection monoids and show how these presentations are compatible with quiver mutations of orientations of Dynkin diagrams with frozen vertices. Our results recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova respectively. Surprisingly, inner by diagram automorphisms of irreducible Weyl groups and Boolean reflection monoids can be constructed by sequences of mutations preserving the same underlying diagrams. Besides, we show that semigroup algebras of Boolean reflection monoids are cellular algebras.
This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them.
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 consequence is a proof of a conjecture of Kubat and Okni{n}ski that every plactic monoid of finite rank satisfies a non-trivial semigroup identity. In the converse direction, we show that every identity satisfied by the plactic monoid of rank $n$ is satisfied by the monoid of $n times n$ upper triangular tropical matrices. In particular this implies that the variety generated by the $3 times 3$ upper triangular tropical matrices coincides with that generated by the plactic monoid of rank $3$, answering another question of Izhakian.
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