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Let R be a ring and X a complex of R-modules. We give some characterizations of the dimension of X related to special Gorenstein projective precovers. Let R be a ring such that (G P , G P ⊥) forms a cotorsion pair cogenerated by a set, where G P denotes the category consisting of all Gorenstein projective R-modules. Then the dimension related to special Gorenstein projective precovers Gppd (X) is equal to the infimum of the set { n ∈ Z | there exists a complete Gorenstein projective resolution of X T → σ G → X such that σi is bijective for each i ≥ n }. Forthmore, if Gppd (X) is finite and n an integer then Gppd (X) ≤ n if and only if Ext G P i (X , Q) = 0 for any i > n and any projective R-module Q, where Ext G P i (− , −) is the relative cohomology functor. [ABSTRACT FROM AUTHOR]

Let R ⊂ S be a Frobenius extension of rings and M an S-complex. It is proven that if the Gorenstein projective (injective) dimension of M as an S-complex is finite, then the Gorenstein projective (injective) dimension of M as an S-complex equals the Gorenstein projective (injective) dimension of M as an R-complex. As a corollary one has that if the projective (injective) dimension of M as an S-complex is finite, then the projective (injective) dimension of M as an S-complex equals the projective (injective) dimension of M as an R-complex. This statement extends a known result of Nakayama and Tsuzuku. [ABSTRACT FROM AUTHOR]

In this paper we consider the monoid D P S n of all partial isometries of a star graph Sn with n vertices. Our main objectives are to determine the rank and to exhibit a presentation of D P S n . We also describe Green's relations of D P S n and calculate its cardinal. [ABSTRACT FROM AUTHOR]

We establish representation types (finite, tame or wild) of finite dimensional Munn algebras with semisimple bases. As an application, we establish representation types of finite Rees matrix semigroups, in particular, 0-simple semigroups, and their mutually annihilating unions. [ABSTRACT FROM AUTHOR]

We introduce the Hochschild extensions of dg algebras, which are A ∞ -algebras. We show that all exact Hochschild extensions are symmetric Hochschild extensions, more precisely, every exact Hochschild extension of a finite dimensional complete typical dg algebra is a symmetric A ∞ -algebra. Moreover, we prove that the Koszul dual of trivial extension is Calabi-Yau completion and the Koszul dual of exact Hochschild extension is deformed Calabi-Yau completion, more precisely, the Koszul dual of the trivial extension of a finite dimensional complete dg algebra is the Calabi-Yau completion of its Koszul dual, and the Koszul dual of an exact Hochschild extension of a finite dimensional complete typical dg algebra is the deformed Calabi-Yau completion of its Koszul dual. [ABSTRACT FROM AUTHOR]

In this article, we study the notion of central extensions of Leibniz n-algebras relative to n-Lie algebras to study properties of Schur Lie -multiplier and Lie -covers on Leibniz n-algebras. We provide a characterization of Lie -perfect Leibniz n-algebras by means of universal Lie -central extensions. It is also provided some inequalities on the dimension of the Schur Lie -multiplier of Leibniz n-algebras. Analogue to Wiegold and Green results on groups or Moneyhun results on Lie algebras, we provide upper bounds for the dimension of the Lie -commutator of a Leibniz n-algebra with finite dimensional Lie -central factor, and also for the dimension of the Schur Lie -multiplier of a finite dimensional Leibniz n-algebra. [ABSTRACT FROM AUTHOR]

Let M n be the variety generated by all monoids of order n. It was known that M n is finitely based if n ⩽ 4 and non-finitely based if n ⩾ 6. However, the finite basis problem for M 5 remained open. In this paper, we show that the variety M 5 is finitely based by giving a finite identity basis. Therefore, M n is finitely based if and only if n ⩽ 5. [ABSTRACT FROM AUTHOR]

We define and study a notion of free wreath product with amalgamation for compact quantum groups. These objects were already introduced in the case of duals of discrete groups under the name "free wreath products of pairs" in a previous work of ours. We give several equivalent descriptions and use them to establish properties like residual finiteness, the Haagerup property or a smash product decomposition. [ABSTRACT FROM AUTHOR]

We introduce the notion of Krull super-dimension of supermodules over certain super-commutative Noetherian super-rings. We investigate how this notion relates to the notion of odd regular sequence due T. Schmitt and how it behaves with respect to the transition to the graded and bigraded super-modules and super-rings associated with the original ones. We also apply these results to the super-dimension theory of superschemes of finite type and their morphisms. [ABSTRACT FROM AUTHOR]

In this article, we introduce the concept of parallel Krull dimension of a module (briefly, p.Krull dimension), which is Krull-like dimension extension of the concept of DCC on poset of submodules parallel to itself. Using this concept, we extend some basic results about modules with this dimension. In particular, we show that an R-module M has Krull dimension if and only if it has homogeneous parallel Krull dimension with finite Goldie dimension and these two dimensions for M coincide. Furthermore, we define the concept of α-DICCP and we prove that M is an α-DICC module if and only if M is an α-DICCP module with finite Goldie dimension. Also, after defining of p-Artinian (resp., p-Noetherian) modules, we prove that if R is semiprime, p-Artinian commutative ring with finite Goldie dimension, then R is a Goldie ring. [ABSTRACT FROM AUTHOR]

Let G be a finite non-abelian p-group of order greater than p4, p an odd prime, such that Z (Ω 1 (H)) is cyclic and Z(H) is elementary abelian, where H = C G (Φ (G)). We prove that the set A (G) = { α ∈ Aut (G) | [ x , α (x) ] = 1 for all x ∈ G } of all commuting automorphisms of G forms a subgroup of Aut (G) if and only if Ω 1 (H) is abelian. Also, we find the structure of A (G) = Aut z (G) for a finite 2-group G of almost maximal class with cyclic center Z(G), where Aut z (G) denotes the set of all central automorphisms of G. [ABSTRACT FROM AUTHOR]

In this article, we classify some finite p-groups whose normal subgroups are characteristic. Particularly we restrict our attention to all finite p-groups of maximal class with an abelian maximal subgroup, finite 3-groups of maximal class, finite split metacyclic p-groups, minimal non-abelian p-groups and all p-groups of maximal class of order less than p6. [ABSTRACT FROM AUTHOR]

Given any group G, the normalizer Hol (G) of the subgroup of left translations in the group of all permutations on G is called the holomorph, and the normalizer NHol (G) of Hol (G) in turn is called the multiple holomorph. The quotient T (G) = NHol (G) / Hol (G) has been computed for various families of groups G in the literature. In this paper, we shall supplement the existing results by considering finite split metacyclic p-groups G with p an odd prime. Our work gives new examples of groups G for which T(G) is not a 2-group. [ABSTRACT FROM AUTHOR]

We consider HNN extensions S * = [ S ; U 1 , U 2 ; ϕ ] where U1 and U2 are inverse monoids of an inverse semigroup S such that, for any u ∈ U i and e ∈ E (S) with u ≥ e in S, there exists f ∈ E (U i) with u ≥ f ≥ e in S, for i ∈ { 1 , 2 } ; we say that U1 and U2 are lower bounded in S. We construct and describe the Schützenberger automata of S * and give conditions for S * to have decidable word problem. Homomorphisms of the Schützenberger graphs of S * are studied and conditions are given for S * to be completely semisimple. When S has decidable word problem and U1 and U2 are finite, we show that S * has decidable word problem. The class of HNN extensions considered here is surprisingly useful and generalizes the class introduced by Jajcayová. A future paper intends to show that any HNN extension of an inverse semigroup can be embedded into an HNN extension where the subsemigroups are lower bounded. [ABSTRACT FROM AUTHOR]

After hereditary rings were defined, as a natural generalization, semi-hereditary rings were introduced. In this article, we study the notion of a finite Σ-Rickart module, as a module theoretic analogue of a right semi-hereditary ring. A module M is called finite Σ-Rickart if every finite direct sum of copies of M is a Rickart module. It is shown that any direct summand and any finite direct sum of copies of a finite Σ-Rickart module are finite Σ-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of semi-hereditary rings: a ring R is right semi-hereditary if and only if every finitely generated submodule of any projective right R-module is projective if and only if every factor module of any injective right R-module is f-injective. Further, we have a characterization of a finite Σ-Rickart module in terms of its endomorphism ring. In addition, we introduce M-coherent modules and provide a characterization of a finite Σ-Rickart module in terms of an M-coherent module. We prove that for any endoregular module M, Hom R (M , A) is an f-injective right S-module for any module A, where S = End R (M). Examples which delineate the concepts and results are provided. [ABSTRACT FROM AUTHOR]

We prove that a finite 3-group in the Cremona group Cr 3 (C) can be generated by at most 4 elements. This provides the last missing piece in bounding the ranks of finite p-subgroups in the space Cremona group. [ABSTRACT FROM AUTHOR]

We determine character tables for twisted fractional linear groups that form the "other" family in Zassenhaus' classification of finite sharply 3-transitive groups. [ABSTRACT FROM AUTHOR]

Given a non-empty set X and let P(X) be a partial transformation semigroup on X. For a fixed subset Y of X, define a generalization of P(X) by PFix (X , Y) = { α ∈ P (X) : y α = y for all y ∈ dom (α) ∩ Y }. In this article, we give a complete description of Green's relations on PFix(X, Y) and apply this to obtain the characterizations of left regular, right regular, intra-regular and completely regular elements of PFix(X, Y). The number and the relationship between these elements are also investigated by using finiteness of X and X ∖ Y. [ABSTRACT FROM AUTHOR]

Let S be a nonempty commutative semigroup written additively. An element e of S is said to be idempotent if e + e = e. The Erdős-Burgess constant I (S) of the semigroup S is defined as the smallest positive integer ℓ such that any S -valued sequence T of length ℓ contains a nonempty subsequence the sum of whose terms is an idempotent of S. We make a study of I (S) when S is a direct product of arbitrarily many of cyclic semigroups. We give the necessary and sufficient conditions such that I (S) is finite, and we obtain sharp bounds of I (S) in case I (S) is finite, and determine the precise value of I (S) in some cases which unifies some well known results on the precise value of Davenport constant in the setting of commutative semigroups. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate certain finite geometric structures, in particular semifields over finite fields, spread-sets, translation planes as a special type of affine planes, and projective planes. Quasifields and their normed spread-sets are also considered. Among other results, the following main results are proved. • If A is a vector semifield a finite field F and B = { f a | 0 ≠ a A } , where f a (x) = x a for all x ∈ A , then A is a semifield if the following holds true: 1. (B ∪ { 0 } , +) is a group and B ⊆ G L (A). 2. The identity map I ∈ B. 3. If a , b ∈ A , a ≠ 0 , then there exists a unique g ∈ B such that a g = b , and conversely if A is a vector space over F and B ⊆ G L (A) satisfying 1 , 2 and 3 then A is a semifield. • If T is the group of translations of the affine plane Π and E ∞ (Π ̂) is the group of elations of the projective plane Π ̂ with axis L ̂ ∞ , then 1. T ≤ Aut (Π) and 2. E ∞ (Π ̂) ≤ Aut (Π ̂) L ̂ ∞ . [ABSTRACT FROM AUTHOR]

We characterize reflectable bases for affine reflection systems (extended affine root systems) of types E 6 , 7 , 8 . This completes the characterization of reflectable bases for reduced affine reflections systems. We prove that a subset of non-isotropic roots is a reflectable base if and only if it is a minimal generating set for the root lattice. Our approach also provides a new less involved proof for simply laced types different from E 6 , 7 , 8 . Using our characterization, we enumerate reflectable bases for the finite type E6. [ABSTRACT FROM AUTHOR]

Two direct-summands A and B of a right R-module M will be called strongly-perspective (s-perspective) if for any X ⊆ M , M = A ⊕ X iff M = B ⊕ X. The module M will be called s-perspective if every two isomorphic direct-summands of M are s-perspective, and a ring R will be called s-perspective if R as a right R-module is s-perspective. In this paper, we study s-perspective rings and modules, and using ideas of Khurana and Nielsen, it follows that if M is s-perspective, then M has the finite exchange property iff M is clean, iff M has the full exchange property. In particular, if M is either summand-square-free or dual-summand-square-free with the finite exchange property, then M is clean, s-perspective and has the full exchange property. [ABSTRACT FROM AUTHOR]

Among finite dimensional algebras over a field K, the class of gentle algebras is known to be closed by derived equivalences. Although a classification up to derived equivalences is usually a difficult problem, Avella-Alaminos and Geiss have introduced derived invariants for gentle algebras A, which can be calculated combinatorially from their bound quivers, applicable to such classification. Ladkani has given a formula to describe the dimensions of the Hochschild cohomologies of A in terms of some values of its Avella-Alaminos–Geiss invariants. This in turn implies a cohomological meaning of these values. Since most of the other values do not appear in this formula, it will be a natural question to ask if there is a similar cohomological meaning for such values. In this article, we construct a sequence of gentle algebras A (k) indexed by positive integers by a procedure which we call finite gentle repetitions, in order to relate these values of Avella-Alaminos–Geiss invariants of A to the dimensions of Hochschild cohomologies of A (k) . On the way we will see that the Avella-Alaminos–Geiss invariants of A (k) are completely determined by those of A. Therefore one may expect that the finite gentle repetitions would preserve derived equivalences in a nice situation. In the last part of this article, we will see how the generalized Auslander–Platzeck–Reiten reflection can be related to the finite gentle repetition. [ABSTRACT FROM AUTHOR]

A group G is subgroup conjugacy distinguished (resp. subgroup into conjugacy separable) if whenever an element y (resp. a finitely generated subgroup K) is not conjugate to an element (to a subgroup) of a finitely generated subgroup H of G there exists a finite quotient G/N of G where yN (resp. KN/N) is not conjugate to an element (to a subgroup) of HN/N. We prove that the fundamental group of a hyperbolic 3-manifold is subgroup conjugacy distinguished and subgroup into conjugacy separable. [ABSTRACT FROM AUTHOR]

In this paper, we first study the species of finite topological spaces recently considered by Fauvet, Foissy, and Manchon. Then, we construct a twisted pre-Lie structure on the species of connected finite topological spaces. The underlying pre-Lie structure defines a coproduct on the species of finite topological spaces different from those already defined by the authors above. In the end, we illustrate the link between the Grossman–Larson product and the proposed coproduct. [ABSTRACT FROM AUTHOR]

Suppose that G is a group, H and K are proper isomorphic central sub-groups of G, and R is an HNN-extension of G with the associated sub-groups H and K. We prove necessary and sufficient conditions for 0 to be residually a C-group, where C is a class of groups closed under taking sub-groups, extensions, homomorphic images, and Cartesian products of the form πyεY where X, Y ε C and Xy is an isomorphic copy of X for each yεY. [ABSTRACT FROM AUTHOR]

Automata over finite alphabet are considered. Initial automata over finite alphabet with 4 states generating amalgamated free products of finite number of finite cyclic groups are constructed. [ABSTRACT FROM AUTHOR]

Xiao and Zhu proved that if C is a locally finite triangulated category, then C has Auslander–Reiten triangles. Let n be a positive integer. The notion of n-abelian categories is a higher analogue of abelian categories. In this paper, we introduce the notion of locally finite n-abelian categories and we prove that they admit n-Auslander–Reiten sequences. As an application, we obtain more existence conditions based on results of Auslander on functor categories. [ABSTRACT FROM AUTHOR]

There are two notions of exponent of finite-dimensional Hopf algebras introduced and studied in the literature. In this note, we discuss and compare their properties including invariance and finiteness. Specifically, one notion is invariant under twisting and taking the Drinfeld double, exactly as the other one. We also find that if the non-cosemisimplicity and dual Chevalley property hold, both exponents are infinite in characteristic 0 but finite in positive characteristic. [ABSTRACT FROM AUTHOR]

Several different generalizations of finite factorization domains (i.e., integral domains where every nonzero nonunit has only finitely many divisors up to associates) have been defined for commutative rings with zero divisors. We study these notions in the context of commutative monoid rings with zero divisors, utilizing semigroup theory to simultaneously generalize and extend many past results about "finite factorization" properties in commutative polynomial rings. Along the way, we expand upon the general theory of factorization in commutative rings with zero divisors, providing new characterizations and results about several kinds of "finite factorization rings." [ABSTRACT FROM AUTHOR]

Let A be a subcategory of an extriangulated category C and let X be a strongly functorially finite subcategory of A. Then the subfactor category A / X admits a pre-triangulated structure in the sense of Beligiannis-Reiten where A has the following properties: (C 2) For any E -triangle A → f B → C − − → in C where f : A → B ∈ A is X -monic, the third term C is also in A ; (C 2 ′) For any E -triangle A → B → g C − − → in C where g : B → C ∈ A is X -epic, the first term A is also in A. Moreover, if the above pre-triangulated category A / X with E (X , X) = 0 becomes a triangulated category, then (A , A) forms an X -mutation pair and A is closed under extensions. [ABSTRACT FROM AUTHOR]

Kumar and Parameswaran introduced formal orbifold curves defined over an algebraically closed field of positive characteristics. They studied both étale and Nori fundamental group schemes associated to such objects. Our aim here is to study the higher dimensional analog of these objects and their fundamental groups. [ABSTRACT FROM AUTHOR]

This paper presents new results on the identities satisfied by the hypoplactic monoid. We show how to embed the hypoplactic monoid of any rank strictly greater than 2 (including infinite rank) into a direct product of copies of the hypoplactic monoid of rank 2. This confirms that all hypoplactic monoids of rank greater than or equal to 2 satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the variety generated by the hypoplactic monoid has finite axiomatic rank, by giving a finite basis for it. [ABSTRACT FROM AUTHOR]

In this article, we describe a criterion for an element of the dual space of an algebra to belong to the finite dual. This result is used to study when a certain subspace of the dual space is a subcoalgebra of the finite dual. We further apply it to find a right alternative coalgebra that is not locally finite. This work is motivated by a conjecture from I. Shestakov, which states that all coalgebras of a given variety are locally finite if, and only if, this variety admits locally nilpotent radical. [ABSTRACT FROM AUTHOR]

In this article, we use quite elementary and simple ideas to rebuild and study the patch and flat topologies on the prime spectrum from a natural point of view (this new approach is based on the significant applications of the power set ring). Especially, the proof of a major result in the literature on the comparison of topologies greatly simplified and shortened. Also a new characterization for the finiteness of the minimal primes of a ring is given. Then as an application, all of the related results of Kaplansky, Anderson, Gilmer-Heinzer, Bahmanpour-Khojali-Naghipour and Naghipour on the finiteness of the minimal primes are easily deduced as special cases of this result. Another finiteness result due to Matlis is also easily obtained which states that a given ring has finitely many minimal primes if and only if no minimal prime is contained in the union of the remaining minimal primes. [ABSTRACT FROM AUTHOR]

Let (R , m) be a complete commutative Noetherian local ring, I an ideal of R, M an R-module (not necessarily I-torsion) and N a finitely generated R-module with Supp R (N) ⊆ V (I) . It is shown that if M is I-ETH-cominimax (i.e. Ext R i (R / I , M) is minimax (or Matlis reflexive), for all i ≥ 0 ) and dim M ≤ 1 or more generally M ∈ FD ≤ 1 , then the R-module Ext R n (M , N) is finitely generated, for all n ≥ 0 . As an application to local cohomology, let Φ be a system of ideals of R and I ∈ Φ , if dim M / a M ≤ 1 (e.g., dim R / a ≤ 1 ) for all a ∈ Φ , then the R-modules Ext R j (H Φ i (M) , N) are finitely generated, for all i ≥ 0 and j ≥ 0 . Similar results are true for local cohomology defined by a pair of ideals and ordinary local cohomology modules. [ABSTRACT FROM AUTHOR]

We give a necessary and sufficient condition for the nonsingular projective toric variety associated to the graph cubeahedron of a finite simple graph to be Fano or weak Fano in terms of the graph. [ABSTRACT FROM AUTHOR]

We give an infinite family of examples that generalize the construction given by Noce, Tracey and Traustason of a locally finite 2-group G containing a left 3-Engel element x where 〈 x 〉 G , the normal closure of x in G, is not nilpotent. The construction is based on a family of Lie algebras that are of interest in their own right and make use of a classical theorem of Lucas, regarding when ( m n ) is even. [ABSTRACT FROM AUTHOR]

D. Prasad showed that the sign of a self-dual representation of a finite or p-adic reductive group is often detected by a central element. We study the extension of his results to some more general situations and make some observations about the consequences of his results. [ABSTRACT FROM AUTHOR]

Let G be a finite group and σ = { σ i | i ∈ I } be a partition of the set of all primes ℙ , that is, ℙ = ∪ i ∈ I σ i and σ i ∩ σ j = ∅ for all i ≠ j. A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi-subgroup of G for some i and H contains exactly one Hall σi-subgroup of G for every σ i ∈ σ (G). A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-subnormal in G if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A t = G such that either A i − 1 is normal in Ai or A i / (A i − 1) A i is σ-primary for all i = 1 , 2 , ... , t ; σ-abnormal in G if K / L K is not σ-primary whenever A ≤ L < K ≤ G and L is a maximal subgroup of K. In this article, we study the structure of a finite group in which σ-primary cyclic subgroups are σ-abnormal or σ-subnormal in G. We also describe the structure of a finite group G which has a complete Hall σ-set { H 1 , H 2 , ... , H t } such that for any A i ≤ H i , G has a subgroup A of order | A 1 | | A 2 | ⋯ | A t | and A is σ-abnormal or σ-subnormal in G. [ABSTRACT FROM AUTHOR]

Let R be a Gorenstein ring. We first study Gorenstein orthogonal classes and Gorenstein cotorsion pairs of R-modules. Then, we introduce the concept of Gorenstein weak tilting R-modules, which is a "Gorenstein analogue" of weak tilting modules. It is proven that a left R-module W is Gorenstein weak n-tilting if and only if its character module W+ is Gorenstein n-cotilting if and only if its Gorenstein flat dimension is at most n and its left Gorenstein Tor-orthogonal class consists precisely of all right R-modules with a G-exact right Prod (W +) -resolution of finite length. Finally, we explore the connections between Gorenstein weak tilting modules and Gorenstein tilting modules. [ABSTRACT FROM AUTHOR]

For each k ≥ 2 , let L k 1 denote the monoid obtained by adjoining an identity element to the semigroup generated by two idempotents E and F subjected to the relation EFEF ⋯ = 0 , where the left side is a product of length k. Under a certain unary operation * , the unary monoid (L k 1 , *) is an involution monoid. By recent results, the monoid L k 1 is non-finitely based if and only if k ≥ 3. As for the involution monoid (L k 1 , *) , it is only known to be non-finitely based for all even k ≥ 2. The objective of this article is to show that (L k 1 , *) is non-finitely based for all odd k ≥ 3. Consequently, (L k 1 , *) is non-finitely based for all k ≥ 2. [ABSTRACT FROM AUTHOR]

Let Λ be an artin algebra and X be a quasi-resolving subcategory of mod - Λ which is of finite type. Let S X (Λ) be the full subcategory of the morphism category H (Λ) consisting of all monomorphisms f : A → B in X such that Coker f also lies in X. In this paper, we state and prove Brauer-Thrall type theorems for S X (Λ) . As applications, we provide necessary and sufficient conditions for the submodule category S (Λ) to be of finite type, whenever Λ is of finite representation type, as well as, for the lower 2 × 2 triangular matrix algebra T 2 (Λ) to be of finite CM-type, whenever Λ is of finite CM-type. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is said to be weakly M -supplemented in G if there exists a subgroup B of G provided that (1) G = HB, and (2) if H 1 / H G is a maximal subgroup of H / H G , then H 1 B = B H 1 < G , where HG is the largest normal subgroup of G contained in H. In this paper, we study the influence of weakly M -supplemented subgroups with given order on the structure of finite groups. [ABSTRACT FROM AUTHOR]

A finite group G is called a PST-group (respectively, PT-group, T-group) if every subnormal subgroup of G is Sylow permutable (respectively, permutable, normal) in G. Let F be a class of group and G a finite group. Then, a set Σ of subgroups of G is called a G-covering subgroup system for the class F if G ∈ F whenever Σ ⊆ F. We prove that: (i) If a set of subgroups Σ of G contains at least one supplement to each maximal subgroup of every Sylow subgroup of G, then, Σ is a G-covering subgroup system for the class of all soluble PST-groups; (ii) if Σ is the set of all two-generated subgroups of G, then, Σ is a G-covering subgroup system for the classes of all soluble PST-groups, all soluble PT-groups, and all soluble T-groups. We use these results to prove the following characterizations of soluble PT-groups and T-groups: Suppose that a set of subgroups Σ contains at least one supplement to each maximal subgroup of every Sylow subgroup of G. Then, G is a soluble PT-group (respectively, a soluble T-group) if and only if every subgroup in Σ is a soluble PT-group (respectively, a soluble T-group) and at least one of the nonidentity Sylow subgroups of G is an Iwasawa (respectively, a Dedekind) group. [ABSTRACT FROM AUTHOR]

A generalized Baumslag–Solitar group (GBS-group) is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. A group is large if some finite index subgroup maps onto the free group F2. An explicit description is given for all index n subgroups in non-large GBS groups. Moreover, we give exact formulas for the numbers of subgroups, normal subgroups, and conjugacy classes of subgroups of each finite index in non-large GBS groups. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe linear groups in which all proper subgroups belong to a group class X for several relevant choices of X. In particular, we study linear groups either with only nilpotent-by-finite proper subgroups or with only paranilpotent subgroups; here a group G is paranilpotent if it has a normal series of finite length whose factors are abelian and have only G-invariant subgroups. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero. By a φ -algebra we mean a superalgebra or an algebra with involution over F. In the last years, the sequence of φ -codimensions { c n φ (A) } n ≥ 1 of a φ -algebra A has been extensively studied. In this paper, we classify varieties generated by unitary φ -algebras having quadratic growth of φ -codimensions. As a consequence we obtain that a unitary φ -algebra with quadratic growth is T φ -equivalent to a finite direct sum of minimal unitary φ -algebras with at most quadratic growth of the φ -codimensions. In addition, we explicit all quadratic functions describing the φ -codimension sequence of a unitary φ -algebra. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of an orientation-preserving transformation on an arbitrary chain, as a natural extension for infinite chains of the well known concept for finite chains introduced in 1998 by McAlister and, independently, in 1999 by Catarino and Higgins. We consider the monoid POP (X) (X) of all orientation-preserving partial transformations on a finite or infinite chain X and its submonoids and POPI (X) of all orientation-preserving full transformations and of all orientation-preserving partial permutations on X, respectively. The monoid OPPO (X) of all order-preserving partial transformations on X and its injective counterpart POI (X) are also considered. We study the regularity and give descriptions of the Green's relations of the monoids POP (X) , PO (X) , OP (X) , POPI (X) , and POI (X). [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional k-algebra and P a 2-term silting complex in K b (proj A). In this article, we investigate the representation dimension of End D b (A) (P) by the silting theory. We show that if P is a separating silting complex with certain homological restriction, then, rep.dim A = rep.dim End D b (A) (P). This result generalizes the earlier result about tilting modules to the silting version. It is well-known that H 0 (P) is a support τ-tilting module. We show that rep.dim End A (H 0 (P)) = rep.dim A / ann A (H 0 (P)) whenever P is both separating and splitting. We apply this to obtain the corresponding consequence for support τ-tilting modules. [ABSTRACT FROM AUTHOR]