Subdirectly irreducible commutative multiplicatively idempotent semirings

Commutative multiplicatively idempotent semirings were studied by the authors and F. ?vr?ek, where the connections to distributive lattices and unitary Boolean rings were established. The variety of these semirings has nice algebraic properties and hence there arose the question to describe this variety, possibly by its subdirectly irreducible members. For the subvariety of so-called Boolean semirings, the subdirectly irreducible members were described by F. Guzmán. He showed that there were just two subdirectly irreducible members, which are the 2-element distributive lattice and the 2-element Boolean ring. We are going to show that although commutative multiplicatively idempotent semirings are at first glance a slight modification of Boolean semirings, for each cardinal n > 1, there exist at least two subdirectly irreducible members of cardinality n and at least 2ⁿ such members if n is infinite. For \({n \in \{2, 3, 4\}}\) the number of subdirectly irreducible members of cardinality n is exactly 2.     

Spectral properties of odd-bipartite <Emphasis Type="Italic">Z</Emphasis>-tensors and their absolute tensors

Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly evenbipartite) tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a Z-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Ztensor. When the order is even and the Z-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the Z-tensor and the largest H-eigenvalue of the absolute tensor of that Z-tensor are equal, if and only if the Z-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Z-tensor with nonnegative diagonal entries and the absolute tensor of the Z-tensor are diagonal similar, if and only if the Z-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric Z-tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the Z-tensor and the spectrum of absolute tensor of that Z-tensor, can be characterized by the equality of their spectral radii.     

The complete reducibility of some <Emphasis Type="Italic">GF</Emphasis>(2)<Emphasis Type="Italic">A</Emphasis><Subscript>7</Subscript>-modules

It is proved that, if G is a finite group with a nontrivial normal 2-subgroup Q such that G/Q ～= A ₇ and an element of order 5 from G acts freely on Q, then the extension G over Q is splittable, Q is an elementary abelian group, and Q is the direct product of minimal normal subgroups of G each of which is isomorphic, as a G/Q-module, to one of the two 4-dimensional irreducible GF(2)A ₇-modules that are conjugate with respect to an outer automorphism of the group A ₇.     

Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle H = （Hu）u∈G^0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle （G^0, （Hu）,μ）, then dimHu = 1 （u ∈ G^0）. Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.     

The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays

Let N = {0, 1, · · ·, n ? 1}. A strongly idempotent self-orthogonal row Latin magic array of order n (SISORLMA(n) for short) based on N is an n × n array M satisfying the following properties: (1) each row of M is a permutation of N, and at least one column is not a permutation of N; (2) the sums of the n numbers in every row and every column are the same; (3) M is orthogonal to its transpose; (4) the main diagonal and the back diagonal of M are 0, 1, · · ·, n ? 1 from left to right. In this paper, it is proved that an SISORLMA(n) exists if and only if n ? {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ? {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠ 2.     

Irreducible <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>1</Subscript><Superscript>(1)</Superscript></Stack>-modules from modules over two-dimensional non-abelian Lie algebra

For any module V over the two-dimensional non-abelian Lie algebra b and scalar α ∈ C, we define a class of weight modules F ^α (V) with zero central charge over the affine Lie algebra A ₁ ⁽¹⁾ . These weight modules have infinitedimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules F ^α(V) to be irreducible. In this way, we obtain a lot of irreducible weight A ₁ ⁽¹⁾ -modules with infinite-dimensional weight spaces.     

Some Properties of the Idempotent Graph of a Ring

The idempotent graph of a ring R, denoted by I(R), is a graph whose vertices are all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = yx = 0. In this paper, we show that diam\({(I(M_n(D))) = 4}\), for all natural number \({n \geq 4}\) and diam\({(I(M_3(D))) = 5}\), where D is a division ring. We also provide some classes of rings whose idempotent graphs are connected. Moreover, the regularity, clique number and chromatic number of idempotent graphs are studied.     

RESTRICTED LIE ALGEBRAS WITH MAXIMAL 0-PIM

In this paper it is shown that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one dimensional trivial module of a maximal torus. As a consequence, the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p ^MT(L), where MT(L) denotes the maximal dimension of a torus in L. Finally, it is proved that in characteristic p > 3 the projective cover of the trivial irreducible L-module is induced from the one-dimensional trivial module of a torus of maximal dimension, only if L is solvable.     

Representation and character varieties of the Baumslag-Solitar groups

Representation and character varieties of the Baumslag–Solitar groups BS(p, q) are analyzed. Irreducible components of these varieties are found, and their dimension is calculated. It is proved that all irreducible components of the representation variety R_n(BS(p, q)) are rational varieties of dimension n², and each irreducible component of the character variety X_n(BS(p, q)) is a rational variety of dimension k ≤ n. The smoothness of irreducible components of the variety R_n^s (BS(p, q)) of irreducible representations is established, and it is proved that all irreducible components of the variety R_n^s (BS(p, q)) are isomorphic to A¹ {0}.     

Cube term blockers without finiteness

We show that an idempotent variety has a d-dimensional cube term if and only if its free algebra on two generators has no d-ary compatible cross. We employ Hall’s Marriage Theorem to show that an idempotent variety \({\mathcal{V}}\) of finite signature whose fundamental operations have arities n ₁, . . . , n _k, has a d-dimensional cube term for some d if and only if it has one of dimension \({1 + \sum_{i=1}^{k} (n_{i} - 1)}\). This upper bound on the dimension of a minimal-dimension cube term for \({\mathcal{V}}\) is shown to be sharp. We show that a pure cyclic term variety has a cube term if and only if it contains no 2- element semilattice. We prove that the Maltsev condition “existence of a cube term” is join prime in the lattice of idempotent Maltsev conditions.     

Certain pairs of irreducible characters of the groups <Emphasis Type="Italic">S</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The investigation of the pairs of irreducible characters of the symmetric group S _n that have the same set of roots in one of the sets A _n and S _n ? A _n is continued. All such pairs of irreducible characters of the group S _n are found in the case when the least of the main diagonal’s lengths of the Young diagrams corresponding to these characters does not exceed 2. Some arguments are obtained for the conjecture that alternating groups A _n have no pairs of semiproportional irreducible characters.     

<Emphasis Type="Italic">M</Emphasis>-slenderness

Analogues of Nunke’s theorem are proved which characterize variants of slenderness. For a bounded monotone subgroup M of ? ^ω , a torsion-free reduced abelian group G is M-slender if, and only if, there is no monomorphism from M into G. It is consistent relative to ordinary set theory (ZFC) that if M ≠ ? ^ω is an unbounded monotone subgroup of ? ^ω , then a torsion-free reduced abelian group G is M-slender if, and only if, there is no monomorphism from M into G.     

Norm estimates of ω-circulant operator matrices and isomorphic operators for ω-circulant algebra

An n ×nω-circulant matrix which has a specific structure is a type of important matrix. Several norm equalities and inequalities are proved for ω-circulant operator matrices with ω = e^iθ (0 ≤ θ < 2π) in this paper. We give the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norms. Pinching type inequality is also proposed for weakly unitarily invariant norms. Meanwhile, we present that the set of ω-circulant matrices with complex entries has an idempotent basis. Based on this basis, we introduce an automorphism on the ω-circulant algebra and then show different operators on linear vector space that are isomorphic to the ω-circulant algebra. The function properties, other idempotent bases and a linear involution are discussed for ω-circulant algebra. These results are closely related to the special structure of ω-circulant matrices.     

Recognition of characteristically simple group <Emphasis Type="Italic">A</Emphasis><Subscript>5</Subscript> × <Emphasis Type="Italic">A</Emphasis><Subscript>5</Subscript> by character degree graph and order

The character degree graph of a finite group G is the graph whose vertices are the prime divisors of the irreducible character degrees of G and two vertices p and q are joined by an edge if pq divides some irreducible character degree of G. It is proved that some simple groups are uniquely determined by their orders and their character degree graphs. But since the character degree graphs of the characteristically simple groups are complete, there are very narrow class of characteristically simple groups which are characterizable by this method.We prove that the characteristically simple group A₅ × A₅ is uniquely determined by its order and its character degree graph. We note that this is the first example of a non simple group which is determined by order and character degree graph. As a consequence of our result we conclude that A₅ × A₅ is uniquely determined by its complex group algebra.     

Higher Algebraic <Emphasis Type="Italic">K</Emphasis>-theory of Ring Epimorphisms

Given a homological ring epimorphism from a ring R to another ring S, we show that if the left R-module S has a finite-type resolution, then the algebraic K-group K _n (R) of R splits as the direct sum of the algebraic K-group K _n (S) of S and the algebraic K-group K _n (R) of a Waldhausen category R determined by the ring epimorphism. This result is then applied to endomorphism rings, matrix subrings, rings with idempotent ideals, and universal localizations which appear often in representation theory and algebraic topology.     

Sets satisfying the Central Sets Theorem

Central subsets of a discrete semigroup S have very strong combinatorial properties which are a consequence of the Central Sets Theorem . We investigate here the class of semigroups that have a subset with zero Følner density which satisfies the conclusion of the Central Sets Theorem. We show that this class includes any direct sum of countably many finite abelian groups as well as any subsemigroup of (?,+) which contains ?. We also show that if S and T are in this class and either both are left cancellative or T has a left identity, then S×T is in this class. We also extend a theorem proved in (Beiglböck et al. in Topology Appl., to appear), which states that, if p is an idempotent in β? whose members have positive density, then every member of p satisfies the Central Sets Theorem. We show that this holds for all commutative semigroups. Finally, we provide a simple elementary proof of the fact that any commutative semigroup satisfies the Strong Følner Condition.     

On the semiproportional character conjecture in groups Sp<Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Previously, the author made the following conjecture: if a finite group has two semiproportional irreducible characters φ and ψ, then φ(1) = ψ(1). In the present paper, a new confirmation of the conjecture is obtained. Namely, the conjecture is verified for symplectic groups Sp₄(q) and PSp₄(q).     

<Emphasis Type="Italic">V</Emphasis>-Gorenstein injective modules preenvelopes and related dimension

Let R and S be associative rings and _S V _R a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom _R (I _V (R),?) and Hom _R (?,I _V (R)) exact exact complex \( \cdots \to {I_1}\xrightarrow{{{d_0}}}{I_0} \to {I^0}\xrightarrow{{{d_0}}}{I^1} \to \cdots \) of V-injective modules I _i and I ⁱ , i ∈ N₀, such that N ? Im(I ₀ → I ⁰). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class A _V (R) which leads to the fact that V-Gorenstein injective modules admit exact right I _V (R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V-Gorenstein injective if and only if N ⊕ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if \(Ext_{{I_V}\left( R \right)}^{ \geqslant n + 1}\left( {I,N} \right) = 0\) for all modules I with finite I _V (R)-injective dimension.     

A new characterization for some extensions of PSL(2, <Emphasis Type="Italic">q)</Emphasis> for some <Emphasis Type="Italic">q</Emphasis> by some character degrees

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p²) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p²) of order 2|PSL(2,p²)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p³. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.     

On the commutation graph of cyclic <Emphasis Type="Italic">TI</Emphasis>-subgroups in linear groups

We study the commutation graph Γ(A) of a cyclic TI-subgroup A of order 4 in a finite group G with quasisimple generalized Fitting subgroup F*(G). It is proved that, if F*(G) is a linear group, then the graph Γ(A) is either a coclique or an edge-regular graph but not a coedge-regular graph.