arXiv subjects

Ivana Đurđev

Publications and source records attributed to Ivana Đurđev.

3 recordsLinked to original sources

arXiv2026

(Super)nilpotency and solvability in algebras with zero

An algebra with zero is an algebra that contains an element 0, called zero, such that all fundamental operations have the property that the result is zero whenever at least one of the arguments takes the value 0. We introduce the notion of the higher commutator of ideals and Rees congruences for such algebras. It is shown that the commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We apply this result to provide characterisations of n-supernilpotency and n-nilpotency in algebras with zero, as well as n-solvability in groupoids with zero.

math.RA↗

arXiv2019

Sandwich semigroups in diagram categories

This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley-Lieb categories. If $\mathcal K$ denotes any of these categories, and if $σ\in\mathcal K_{nm}$ is a fixed morphism, then an associative operation $\star_σ$ may be defined on $\mathcal K_{mn}$ by $α\star_σβ=ασβ$. The resulting semigroup $\mathcal K_{mn}^σ=(\mathcal K_{mn},\star_σ)$ is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green's relations and preorders, regularity, stability, mid-identities, ideal structure, (products of) idempotents, and minimal generation. It turns out that the Brauer category has many remarkable properties not shared by any of the other diagram categories we study. Because of these unique properties, we may completely classify isomorphism classes of sandwich semigroups in the Brauer category, calculate the rank (smallest size of a generating set) of an arbitrary sandwich semigroup, enumerate Green's classes and idempotents, and calculate ranks (and idempotent ranks, where appropriate) of the regular subsemigroup and its ideals, as well as the idempotent-generated subsemigroup. Several illustrative examples are considered throughout, partly to demonstrate the sometimes-subtle differences between the various diagram categories.

math.GR↗

arXiv2018

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 $i\to j$. 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 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 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 $S_{ij}^a$, Reg$(S_{ij}^a)$ and $\mathbb E(S_{ij}^a)$; we give lower bounds for these ranks, and in the case of 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.

math.GR↗