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If $\mathcal{C}$ is a category of algebras closed under finite direct products, and $M_\mathcal{C}$ the commutative monoid of isomorphism classes of members of $\mathcal{C},$ with operation induced by direct product, A.Tarski defined a…

We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which seem to be necessary to make the…

We initiate the study of the expansion $\mathcal{S}(M)$ of a monoid $M$ obtained via the semidirect product of $M$ acting naturally on the left of its power set (regarded as a semilattice under union). We term this the `subset expansion' of…

For each positive $n$, let $u_n = v_n$ denote the identity obtained from the Adjan identity $(xy) (yx) (xy) (xy) (yx) = (xy) (yx) (yx) (xy) (yx)$ by substituting $(xy) \rightarrow (x_1 x_2 \dots x_n)$ and $(yx) \rightarrow (x_n \dots x_2…

In this paper I consider the structure of the polylinear mapping of the free algebra over the commutative ring.

This work creates two categories of "array-weighted sets" for the purposes of constructing universal matrix-normed spaces and algebras. These universal objects have the analogous universal property to the free vector space, lifting maps…

The main result of the paper proves that the monoid of all 2 x 2 tropical matrices satisfies a non-trivial semigroup identity, unlike the case of matrices over infinite fields. We exploit a strong connection between the monoid of all 2 x 2…

In this paper, we introduce and investigate \emph{semicorings} over associative semirings and their categories of \emph{semicomodules.} Our results generalize old and recent results on corings over rings and their categories of comodules.…

We show that the set of all (unimodular and non-unimodular) free cyclic submodules of T^2, where T is the ring of ternions over a commutative field, admits a point model in terms of a smooth algebraic variety.

We examine two associative products over the ring of symmetric functions related to the intransitive and Cartesian products of permutation groups. As an application, we give an enumeration of some Feynman type diagrams arising in Bender's…

We prove that triangular configurations are plentiful in large subsets of cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is…

Some varieties of groupoids and quasigroups generated by linear-bivariate polynomials $P(x,y)=a+bx+cy$ over the ring $\mathbb{Z}_n$ are studied. Necessary and sufficient conditions for such groupoids and quasigroups to obey identities which…

We have shown recently that, given a metric space $X$, the coarse equivalence classes of metrics on the two copies of $X$ form an inverse semigroup $M(X)$. Here we study the property of idempotents in $M(X)$ of being finite or infinite,…

We consider certain decision problems for the free model of the theory of Cartesian monoids. We introduce a model of computation based on the notion of a single stack one-way PDA due to Ginsburg, Greibach and Harrison. This model allows us…

This is a contribution to the project of quiver approaches to quasi-quantum groups initiated in arXiv:0902.1620. We classify Majid bimodules over groups with 3-cocycles by virtue of projective representations. This leads to a theoretic…

Let Q be a Dynkin quiver of type A. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. We classify the maximal rigid…

We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of…

This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X of S, decide whether each element of S has at most one factorization over X. To…

Let $K$ be a field, let $\sigma$ be an automorphism of $K$, and let $\delta$ be a derivation of $K$. We show that if $D$ is one of $K(x;\sigma)$ or $K(x;\delta)$, then $D$ either contains a free algebra over its center on two generators, or…

We investigate invertible projective representations and their 2-categorical analogues using the language of TQFTs with defects. The main result is a freeness property for invertible projective representatios. While trivial in the…