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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.
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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.
We exhibit faithful representations of the hypoplactic, stalactic, taiga, sylvester, Baxter and right patience sorting monoids of each finite rank as monoids of upper triangular matrices over any semiring from a large class including the tropical semiring and fields of characteristic $0$. By analysing the image of these representations, we show that the variety generated by a single hypoplactic (respectively, stalactic or taiga) monoid of rank at least $2$ coincides with the variety generated by the natural numbers together with a fixed finite monoid $mathcal{H}$ (respectively, $mathcal{F}$) forming a proper subvariety of the variety generated by the plactic monoid of rank $2$.
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 build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyals category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices.
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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