Semigroup varieties with a small number of term operations

We give an equational characterization of (varieties of) semigroups having a p_n-sequence bounded above by a polynomial function of n. This is achieved by studying the syntactical connections between certain semigroup identities and their equational consequences.     

nvolutorial Płonka sums

The construction of the sum of a direct (semilattice ordered) system of algebras introduced by J. Plonka – later known as the Plonka sum – is one of the most important methods of composition in universal algebra, having a number of applications in different algebraic theories, such as semigroup theory, semiring theory, etc. In this paper we present a more general way for constructing algebras with involution, that is, algebraic systems equipped with a unary involutorial operation which is at the same time an antiautomorphism of the underlying algebra. It is the sum – involutorial Plonka sum, as we call it – of an involution semilattice ordered system of algebras. We investigate its basic properties, as well as the problem of its subdirect decomposition.     

Quasilinear varieties of semigroups

We characterize all quasilinear varieties of semigroups, i.e. semigroup varieties V{\mathcal{V}} with the property that for each word w there is a linear word w′ such that V{\mathcal{V}} satisfies w≈w′.     

A remark on nonfinitely based semirings

We generalize a criterion from a previous paper which ensures that an additively idempotent semiring is not finitely based. As a consequence, we prove the NFB property for the semiring generated by transformations on a finite set with more than one element. Supported by Grant No. 144011 of the Ministry of Science of the Republic of Serbia.     

A conference report: The 5th Novi Sad Algebraic Conference (in conjunction with AAA94)

Headlining the Topical Collection dedicated to The 5th Novi Sad Algebraic Conference (NSAC 2017), we provide a brief report of the conference along with some of its history and background.     

Sandwich semigroups in locally small categories I: foundations

Fix (not necessarily distinct) objects i and j of a locally small category S, and write \(S_{ij}\) for the set of all morphisms \(i\rightarrow j\). Fix a morphism \(a\in S_{ji}\), and define an operation \(\star _a\) on \(S_{ij}\) by \(x\star _ay=xay\) for all \(x,y\in S_{ij}\). Then \((S_{ij},\star _a)\) is a semigroup, known as a sandwich semigroup, and denoted by \(S_{ij}^a\). This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green’s relations and stability, focusing on the relationships between these properties on \(S_{ij}^a\) and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set \({\text {Reg}}(S_{ij}^a)\) of all regular elements of \(S_{ij}^a\) is a subsemigroup of \(S_{ij}^a\). Under this condition, we carefully analyse the structure of the semigroup \({\text {Reg}}(S_{ij}^a)\), relating it via pullback products to certain regular subsemigroups of \(S_{ii}\) and \(S_{jj}\), and to a certain regular sandwich monoid defined on a subset of \(S_{ji}\); among other things, this allows us to also describe the idempotent-generated subsemigroup \(\mathbb E(S_{ij}^a)\) of \(S_{ij}^a\). We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups \(S_{ij}^a\), \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\); we give lower bounds for these ranks, and in the case of \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\) show that the bounds are sharp under a certain condition we call MI-domination. Applications to concrete categories of transformations and partial transformations are given in Part II.     

A class of inherently nonfinitely based semirings

We give a sufficient condition which ensures that a semiring with an idempotent addition is inherently nonfinitely based. This enables us to provide a number of small and natural examples of nonfinitely based semirings, including semirings of binary relations on a finite set. Supported by Grant No.144011 of the Ministry of Science of the Republic of Serbia.