Algebra Colloquium

Let G be a universal triangle group, and H a subgroup of G such that the chamber system Δ_H is a tight triangle geometry. Then H, which is canonically isomorphic to the topological fundamental group π₁(Δ_H) of Δ_H, is a finitely presented group. For some H we give their representations.

P-Compatibility is a hereditary property of identities which generalizes the properties of normality and externality of identities. Chajda characterized the normalization of a variety by an algebraic construction called a choice algebra. In this paper, we generalize this characterization to the least P-compatible variety P(V) determined by a variety V for any partition P using P-choice algebras. We also study the clone of (strongly) P-compatible n-ary terms of a variety V, and relate identities of this clone to (strongly) P-compatible hyperidentities of the variety V.

It has been conjectured that if the number of distinct irreducible constituents of the product of two faithful irreducible characters of a finite p-group, for p ≥ 5, is bigger than (p + 1)/2, then it is at least p. We give a counterexample to this conjecture.

For a finitely generated module M over a commutative Noetherian local ring (R,𝔪), it is shown that there exist only a finite number of non-isomorphic top local cohomology modules for all ideals 𝔞 of R. It is also shown that for a given integer r ≥ 0, if is zero for all 𝔭 in Supp(M), then for all i ≥ r.

In this paper, we study the structure of a superabundant semigroup S whose set of idempotents E(S) forms a subsemigroup. We call such a semigroup a cyber-group because it is a generalization of orthogroups in the class of completely regular semigroups studied by Petrich and Reilly. We show that a cyber-group can be expressed as a semilattice of rectangular monoids. Thus, our result generalizes the well-known result obtained by Petrich in 1987 for orthogroups. Some properties of cyber-groups are given and some special superabundant semigroups are discussed.

For a weak entwining structure (A, C, ψ) living in a braided monoidal category with equalizers and coequalizers, we formulate the notion of weak A-Galois coextension with normal basis and we show that these Galois coextensions are equivalent to the weak A-cocleft coextensions introduced by the authors.

In this paper, we prove that a completely 0-simple (or completely simple) semigroup is an amalgamation base for finite semigroups if and only if it is an amalgamation base for semigroups. By adopting the same method as used in a previous paper, we prove that a finite regular semigroup is an amalgamation base for finite semigroups if its -classes are linearly ordered and all of its principal factor semigroups are amalgamation bases for finite semigroups. Finally, we give an example of a finite semigroup U which is an amalgamation base for semigroups, but not all of its principal factor semigroups are amalgamation bases either for semigroups or for finite semigroups.

In 1976 Howie proved that a finite congruence-free semigroup is a simple group if it has at least three elements but no zero element. Infinite congruence-free semigroups are far more complicated to describe, but some have been constructed using semigroups of transformations (for example, by Howie in 1981 and by Marques in 1983). Here, for certain semigroups S of numbers and of transformations, we determine all congruences ρ on S such that S/ρ is congruence-free, that is, we describe all maximal congruences on such semigroups S.

We study two classes of Artinian modules called co-Buchsbaum modules and generalized co-Cohen–Macaulay modules. Some basic properties and characterizations of these modules in terms of 𝔮-weak co-sequences, co-standard sequences, multiplicity, local homology modules are presented.

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

The concept of wreath product of semigroups was initiated by Neumann in 1960, and later on, his concept was used by Preston to investigate the structure of some inverse semigroups. Recently, we start to investigate the structure of left C-rpp semigroups by using wreath products. In this paper, we modify the wreath product to “dual wreath product” so that we can study the structure of right C-rpp semigroups. We prove that a semigroup is a right C-rpp semigroup if and only if it is the dual wreath product of a right regular band and a C-rpp semigroup. Our theorem provides new insight to the structure of right C-rpp semigroups. In particular, a recent result given by Ren and Shum for right C-rpp semigroups is strengthened and enriched.

On the boundary of a polyhedron, we construct a tessellation consisting of 8n pentagons (n ≥ 3). The tessellation produces a family of 3-dimensional orientable closed manifolds with a spine, which correspond to group presentations. We determine the group presentations and their homology groups.

We introduce the graded bialgebra deformations, which explain the lifting method of Andruskiewitsch and Schneider. We also relate these graded bialgebra deformations with the corresponding graded bialgebra cohomology groups, which is the graded version of the one due to Gerstenhaber and Schack.

In this paper, we classify the nonassociative algebras whose associator satisfies relations defined by a natural action of the symmetric group of degree 3.

Let M_R be a right R-module over a ring R with S = End(M_R). We study the coherence of the left S-module _SM relative to a torsion theory for the category of right R-modules. Various results are developed, many extending known results.

Lee weight is more appropriate for some practical situations than Hamming weight as it takes into account the magnitude of each digit of the word. In this paper, we obtain a sufficient bound over the number of parity check digits for codes detecting burst errors and also for codes correcting burst errors with Lee weight.

For a prime p, let D_4p be the dihedral group 〈a,b | a^2p = b² = 1, b^-1ab = a^-1〉 of order 4p, and Cay(G,S) a connected cubic Cayley graph of order 4p. In this paper, it is shown that the automorphism group Aut(Cay(G,S)) of Cay(G,S) is the semiproduct R(G) ⋊ Aut(G,S), where R(G) is the right regular representation of G and Aut(G,S) = {α ∈ Aut(G) | S^α = S}, except either G = D_4p (p ≥ 3), S^β = {b,ab,a^pb} for some β ∈ Aut(D_4p) and , or Cay(G,S) is isomorphic to the three-dimensional hypercube Q₃ and G = ℤ₄ × ℤ₂ or D₈.