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In this paper, a question due to Heckenberger, Shareshian and Welker on racks in Heckenberger et al. (Trans Am Math Soc, 372:1407–1427, 2019) is positively answered. A rack is a set together with a self-distributive bijective binary operation. We show that the lattice of subracks of every finite rack is complemented. Moreover, we characterize finite modular lattices of subracks in terms of complements of subracks. Also, we introduce a certain class of racks including all finite groups with the conjugation operation, called G-racks, and we study some of their properties. In particular, we show that a finite G-rack has the homotopy type of a sphere. Further, we show that the lattice of subracks of an infinite rack is not necessarily complemented which gives an affirmative answer to the aforementioned question. Indeed, we show that the lattice of subracks of the set of rational numbers, as a dihedral rack, is not complemented. Finally, we show that being a Boolean algebra, pseudocomplemented and uniquely complemented as well as distributivity are equivalent for the lattice of subracks of a rack.

In this paper we prove the conjectured upper bound for Castelnuovo–Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the number of maximal cliques of G, denoted by c(G). Moreover, we classify all chordal graphs G for which $$\mathcal {L}(G)=c(G)$$ , where $$\mathcal {L}(G)$$ is the sum of the lengths of longest induced paths of connected components of G. We call such graphs strong interval graphs. We show that the regularity of a strong interval graph G coincides with $$\mathcal {L}(G)$$ as well as c(G).

In this paper, we present some recent instances of applying algebraic tools from different categories, in the design of digital communications systems. More specifically, we present a structure based on the elements of a group ring for constructing and encoding QC-LDPC codes. As another instance, we present the construction of space-time block codes from the rings of twisted Laurent series which include some instances of crossed product and non-crossed product division algebras.

A graph $$G=(V(G), E(G))$$ is supereulerian if it has a spanning Eulerian subgraph. Let $$\ell (G)$$ be the maximum number of edges of spanning Eulerian subgraphs of a graph G. Motivated by a conjecture due to Catlin on supereulerian graphs, it was shown that if G is an r-regular supereulerian graph, then $$\ell (G)\ge \frac{2}{3}|E(G)|$$ when $$r\ne 5$$ , and $$\ell (G)> \frac{3}{5}|E(G)|$$ when $$r=5$$ . In this paper we improve the coefficient and prove that if G is a 5-regular supereulerian graph, then $$\ell (G)\ge \frac{19}{30}|E(G)|+\frac{4}{3}$$ . For this, we first show that each graph G with maximum degree at most 3 has a matching with at least $$\frac{2}{7}|E(G)|$$ edges and this bound is sharp. Moreover, we show that Catlin’s conjecture holds for claw-free graphs having no vertex of degree 4. Indeed, Catlin’s conjecture does not hold for claw-free graphs in general.

Let $$G$$ G be a graph of order $$n$$ n and size $$m$$ m . Suppose that $$f:V(G)\rightarrow {\mathbb {N}}$$ f : V ( G ) ? N is a function such that $$\sum _{v\in V(G)}f(v)=m+n$$ ? v ? V ( G ) f ( v ) = m + n . In this paper we provide a criterion for $$f$$ f -choosability of $$G$$ G . Using this criterion, it is shown that the choice number of the complete $$k$$ k -partite graph $$K_{2,2,\ldots ,2}$$ K 2 , 2 , ? , 2 is $$k$$ k , which is a well-known result due to Erdos, Rubin and Taylor. Among other results we study the $$f$$ f -choosability of the complete $$k$$ k -partite graphs with part sizes at most $$2$$ 2 , when $$f(v)\in \{k-1,k\}$$ f ( v ) ? { k - 1 , k } , for every vertex $$v\in V(G)$$ v ? V ( G ) .

Let D be an infinite division ring. A famous result due to Herstein says that every non-central element of D has infinitely many conjugates and so, if D * is an FC-group, then D is a field. Let M be a maximal subgroup of GL n (D), where n ≥ 1. In this paper, we prove that if M is an FC-group, then it is the multiplicative group of some maximal subfield of M n (D). Moreover, if M is algebraic over Z(D), then [D : Z(D)]

Suppose that D is a division ring with center F and N is a non-central normal subgroup of GLn(D). In this paper we generalize some known results about maximal subgroups of GLn(D) to maximal subgroups of N. More precisely, we prove that if M is a maximal subgroup of N such that F[M] satisfies a polynomial identity and \(C_{M_{n}(D)}(M) \backslash F\) contains an algebraic element over F or \(C_{M_{n}(D)}(M)=F\) and either n ≥ 2 or n = 1 and M is not abelian, then [D : F] < ∞.

LetD be a finite-dimensionalF-central division algebra. A criterion is given forD to be a supersoluble (nilpotent) crossed product division algebra in terms of subgroups of the multiplicative groupD* ofD. More precisely, it is shown thatD is a supersoluble (nilpotent) crossed product if and only ifD* contains an abelian-by-supersoluble (abelian-by-nilpotent) generating subgroup.