Narrow Results

We prove that the Brinkmann Problems (BrP & BrCP) and the Twisted-Conjugacy Problem (TCP) are decidable for any endomorphism of a free-abelian times free (FATF) group ${𝔽}_n\times {\mathbb{Z}}^m$. Furthermore, we prove the decidability of the two-sided Brinkmann Conjugacy Problem (2BrCP) for monomorphisms of FATF groups (and combine it with TCP) to derive the decidability of the conjugacy problem for ascending HNN extensions of FATF groups.

The premise of the soft set was designed to model vague ideas. In this kind of model, where the set of parameters exhibits some degree of order, lattice order theory is quite beneficial. For researchers studying uncertainty, the notion of soft sets, lattices, and fuzzy sets has proven essential. In this study, the postulation of the Lattice ordered Linear Diophantine Multi-Fuzzy Soft Set (LLDMFSS) is laid out. The fundamental objective of this study is the formation of hybrid algebraic structures for LLDMFSS. Particularly, the characteristics of complete lattice, modular lattice, and distributive lattice were inhibited in LLDMFSS, and some pertinent findings were obtained. Subsequently, the modular and distributive LLDMFSS characterization theorem was implemented. The notion of the direct product for two LLDMFSSs was then addressed.

The purpose of this paper is to introduce the notion of several kinds of products of rough transformation semigroups. We establish relationship through coverings and investigate some algebraic properties with regard to these products.

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. An edge subset $F\subseteq E(G)$ is called a restricted edge-cut if $G-F$ is disconnected and has no isolated vertices. The restricted edge-connectivity ${\lambda }^{\prime }(G)$ of $G$ is the cardinality of a minimum restricted edge-cut of $G$ if it has any; otherwise ${\lambda }^{\prime }(G)=+\infty$. If $G$ is not a star and its order is at least four, then ${\lambda }^{\prime }(G)\le \xi (G)$, where $\xi (G)=min\{d_G(u)+d_G(v)-2:uv\in E(G)\}$. The graph $G$ is said to be maximally restricted edge-connected if ${\lambda }^{\prime }(G)=\xi (G)$; the graph $G$ is said to be super restricted edge-connected if every minimum restricted edge-cut isolates an edge from $G$. The direct product of graphs $G_1$ and $G_2$, denoted by $G_1\times G_2$, is the graph with vertex set $V(G_1\times G_2)=V(G_1)\times V(G_2)$, where two vertices $(u_1,v_1)$ and $(u_2,v_2)$ are adjacent in $G_1\times G_2$ if and only if $u_1{u}_2\in E(G_1)$ and $v_1{v}_2\in E(G_2)$. In this paper, we give a sufficient condition for $G\times K_n$ to be super restricted edge-connected, where $K_n$ is the complete graph on $n$ vertices.

We give conditions under which a quasi-isometric map between direct products of hyperbolic spaces splits as a direct product up to bounded distance and permutation of factors. This is a variation on a result due to Kapovich, Kleiner and Leeb.

We solve the subgroup intersection problem (SIP) for any RAAG $G$ of Droms type (i.e. with defining graph not containing induced squares or paths of length $3$): there is an algorithm which, given finite sets of generators for two subgroups $H,K\le G$, decides whether $H\cap K$ is finitely generated or not, and, in the affirmative case, it computes a set of generators for $H\cap K$. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. ${𝔽}_2\times {𝔽}_2$) even have unsolvable SIP.

A semigroup $S$ is right noetherian if every right congruence on $S$ is finitely generated. In this paper, we present some fundamental properties of right noetherian semigroups, discuss how semigroups relate to their substructures with regard to the property of being right noetherian, and investigate whether this property is preserved under various semigroup constructions.

A linear k-forest is a forest whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted by ${\text{la}}_k(G)$, is the least number of linear k-forests needed to decompose G. Recently, Zuo, He, and Xue studied the exact values of the linear $(n-1)$-arboricity of Cartesian products of various combinations of complete graphs, cycles, complete multipartite graphs. In this paper, for general k we show that $\max \{{\text{la}}_k(G),{\text{la}}_l(H)\}\le {\text{la}}_{\max \{k,l\}}(G\square H)\le {\text{la}}_k(G)+{\text{la}}_l(H)$ for any two graphs G and H. Denote by $G\circ H,G\times H$ and $G⊠H$ the lexicographic product, direct product and strong product of two graphs G and H, respectively. For any two graphs G and H, we also derive upper and lower bounds of ${\text{la}}_k(G\circ H),{\text{la}}_k(G\times H)$ and ${\text{la}}_k(G⊠H)$ in this paper. The linear k-arboricity of a 2-dimensional grid graph, a r-dimensional mesh, a r-dimensional torus, a r-dimensional generalized hypercube and a hyper Petersen network are also studied.

We determine entirely which Artinian rings have maximal subring. In particular, we show that an Artinian ring without maximal subring is integral over some finite subring and in particular that every Artinian ring which is uncountable or of characteristic zero has a maximal subring. We also determine when a finite direct product of rings has a maximal subring. Finally, we show that if a ring R has an Artinian maximal subring then R itself is Artinian.

Let R be a ring with identity. A unital left R-module M has the min-property provided the simple submodules of M are independent. On the other hand a left R-module M has the complete max-property provided the maximal submodules of M are completely coindependent, in other words every maximal submodule of M does not contain the intersection of the other maximal submodules of M. A semisimple module X has the min-property if and only if X does not contain distinct isomorphic simple submodules and this occurs if and only if X has the complete max-property. A left R-module M has the max-property if for every positive integer n and distinct maximal submodules L, L_i (1 ≤ i ≤ n) of M. It is proved that a left R-module M has the complete max-property if and only if M has the max-property and every maximal submodule of M/Rad M is a direct summand, where Rad M denotes the radical of M, and in this case every maximal submodule of M is fully invariant. Various characterizations are given for when a module M has the max-property and when M has the complete max-property.

Let $\mathrm{\text{cd}}(G)$ be the set of all irreducible complex characters of a finite group $G$. In [K. Aziziheris, Determining group structure from sets of irreducible character degrees, J. Algebra323 (2010) 1765–1782], we proved that if $m$ and $n$ are relatively prime integers greater than $1$, $p$ is prime not dividing $mn$, and $G$ is a solvable group such that $\mathrm{\text{cd}}(G)=\{1,p,n,m,pn,pm\}$, then under some conditions on $p,m,$ and $n$, the group $G=A\times B$ is the direct product of two normal subgroups, where $\mathrm{\text{cd}}(A)=\{1,p\}$ and $\mathrm{\text{cd}}(B)=\{1,n,m\}$. In this paper, we replace $p$ by $u$, where $u>1$ is an arbitrary positive integer, and we obtain similar result. As an application, we show that if $G$ is a finite group with $\mathrm{\text{cd}}(G)=\{1,21,13,55,273,1155\}$ or $\mathrm{\text{cd}}(G)=\{1,15,391,58,5865,870\}$, then $G$ is a direct product of two non-abelian normal subgroups.

In this paper, we define an equivalence relation induced by $n$-ary subpolygroups and show that such relation is full conjugation when $n$-ary subpolygroup is normal. Then, we introduce a construction and some properties of direct product of $n$-ary polygroups via $n$-ary factor polygroups. This construction in principle is enlargement of elements of $n$-ary polygroup except zero. Finally, we introduce and investigate the semidirect product of $n$-ary polygroups via $n$-ary factor polygroups.

Given any minimal ring extension $k\subset L$ of finite fields, several families of examples are constructed of a finite local (commutative unital) ring $A$ which is not a field, with a (necessarily finite) inert (minimal ring) extension $A\subset B$ (so that $B$ is a separable $A$-algebra), such that $A\subset B$ is not a Galois extension and the residue field of $A$ (respectively, $B$) is $k$ (respectively, $L$). These results refute an assertion of G. Ganske and McDonald stating that if $R\subseteq S$ are finite local rings such that $S$ is a separable $R$-algebra, then $R\subseteq S$ is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let $(A,M)$ be a finite special principal ideal ring (SPIR), but not a field, such that $M$ has index of nilpotency $\alpha$ ($\ge 2$). Impose the uniform distribution on the (finite) set of ($A$-algebra) isomorphism classes of the minimal ring extensions of $A$. If $2\in M$ (for instance, if $A\cong \mathbb{Z}/2^{\alpha }\mathbb{Z}$), the probability that a random isomorphism class consists of ramified extensions of $A$ is at least $2/3$; if $2\notin M$ (for instance, if $A\cong \mathbb{Z}/p^{\alpha }\mathbb{Z}$ for some odd prime $p$), the corresponding probability is at least $3/4$. Additional applications, examples and historical remarks are given.

A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $〈H,H^g〉$ for each $g\in G$. Some problems in Finite Group Theory, Combinatorics and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper, we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd index in some direct products of finite groups. In particular, in this paper, we prove that the subgroups of odd index are pronormal in the direct product $G$ of finite simple symplectic groups over fields of odd characteristics if and only if the subgroups of odd index are pronormal in each direct factor of $G$. Moreover, deciding the pronormality of a given subgroup of odd index in the direct product of simple symplectic groups over fields of odd characteristics is reducible to deciding the pronormality of some subgroup $H$ of odd index in a subgroup of ${\prod }_{i=1}^t{\mathbb{Z}}_3\wr {\mathrm{\text{Sym}}}_{n_i}$, where each ${\mathrm{\text{Sym}}}_{n_i}$ acts naturally on $\{1,\dots ,n_i\}$, such that $H$ projects onto ${\prod }_{i=1}^t{\mathrm{\text{Sym}}}_{n_i}$. Thus, in this paper, we obtain a criterion of pronormality of a subgroup $H$ of odd index in a subgroup of ${\prod }_{i=1}^t{\mathbb{Z}}_{p_i}\wr {\mathrm{\text{Sym}}}_{n_i}$, where each $p_i$ is a prime and each ${\mathrm{\text{Sym}}}_{n_i}$ acts naturally on $\{1,\dots ,n_i\}$, such that $H$ projects onto ${\prod }_{i=1}^t{\mathrm{\text{Sym}}}_{n_i}$.

Partially ordered monoids (or pomonoids) $S$ acting on a partially ordered set (or poset), briefly $S$-posets, appear naturally in the study of mappings between posets, and play an essential role in pomonoid theory. The study of flatness properties of $S$-posets was initiated by Fakhruddin in the 1980s, and an extensive theory of flatness properties has been developed in the past several decades. The obtained results have prompted a new progress in the research area of $S$-posets. To date, a large number of familiar properties have been generalized from acts to $S$-posets (involving free and projective $S$-posets, flat $S$-posets of various sorts, $S$-posets satisfying Conditions $(P)$, $(WP)$ and $(PWP)$, and torsion free $S$-posets). Some new properties in $S$-posets, such as Conditions $(P_w)$, ${(WP)}_w$ and ${(PWP)}_w$, have also been discovered. This paper continues the study of flatness properties of $S$-posets. We first introduce Condition $(P^{\prime })$ in the context of $S$-posets, and characterize pomonoids by this new property of $S$-posets. Unlike the case for acts, pomonoids over which all right $S$-posets satisfy Condition $(P^{\prime })$ are stronger than pogroups. Thereby, we introduce Condition $(P_w^{\prime })$ similar to Condition $(P_w)$. Furthermore, we describe Conditions $(P^{\prime })$ and $(P_w^{\prime })$ covers of cyclic $S$-posets. Finally, we investigate direct products of $S$-posets satisfying Conditions $(P^{\prime })$ and $(P_w^{\prime })$.

The groupoid description of Schwinger’s picture of quantum mechanics is continued by discussing the closely related notions of composition of systems, subsystems, and their independence. Physical subsystems have a neat algebraic description as subgroupoids of the Schwinger’s groupoid of the system. The groupoid picture offers two natural notions of composition of systems: Direct and free products of groupoids, that will be analyzed in depth as well as their universal character. Finally, the notion of independence of subsystems will be reviewed, finding that the usual notion of independence, as well as the notion of free independence, find a natural realm in the groupoid formalism. The ideas described in this paper will be illustrated by using the EPRB experiment. It will be observed that, in addition to the notion of the non-separability provided by the entangled state of the system, there is an intrinsic “non-separability” associated to the impossibility of identifying the entangled particles as subsystems of the total system.

We give necessary and sufficient conditions for the efficiency of the direct product of finitely many finite monogenic monoids.

In this paper, we introduce the concept of a strongly ordered congruence on a directed ordered semigroup S. We prove that any strongly ordered congruence on S is a strongly regular congruence. We characterize the finite direct product, subdirect product and full subdirect product of ordered semigroups by using the concepts of strongly ordered congruence and regular congruence on an ordered semigroup S.

In this paper, we introduce power-serieswise McCoy rings, which are a generalization of power-serieswise Armendariz rings, and investigate their properties. We show that a ring R is power-serieswise McCoy if and only if the ring consisting of n × n upper triangular matrices with equal diagonal entries over R is power-serieswise McCoy. We also prove that a direct product of rings is power-serieswise McCoy if and only if each of its factors is power-serieswise McCoy. Meanwhile we show that power-serieswise McCoy rings may be neither semi-commutative nor power-serieswise Armendariz.

It is proved that a decomposition of a weak BCC-algebra X into branches induces on X a congruence θ such that the algebra X/θ is isomorphic to the subalgebra of X containing all initial elements of X. A decomposition of a medial weak BCC-algebra into a direct product of subalgebras generated by one element is also presented.