A Note on Semigroup Algebras of Permutable Semigroups

Let S be a semigroup and 𝔽 be a field. For an ideal J of the semigroup algebra 𝔽[S] of S over 𝔽, let ?_J denote the restriction (to S) of the congruence on 𝔽[S] defined by the ideal J. A semigroup S is called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β of S. In this paper we show that if S is a semilattice or a rectangular band then φ_{{S; 𝔽}}: J → ?_J is a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (?_S; ○ ) if and only if S is a permutable semigroup.     

Semiprimitivity of Orthodox Semigroup Algebras

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R₀[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.     

On finiteness conditions for Rees matrix semigroups

Let T=[S; I; J; P] be a Rees matrix semigroup where S is a semigroup, I and J are index sets, and P is a J × I matrix with entries from S, and let U be the ideal generated by all the entries of P. If U has finite index in S, then we prove that T is periodic (locally finite) if and only if S is periodic (locally finite). Moreover, residual finiteness and having solvable word problem are investigated.     

On the Jacobson Radical of Skew Polynomial Extensions of Rings Satisfying a Polynomial Identity

Let R be a ring satisfying a polynomial identity, and let D be a derivation of R. We consider the Jacobson radical of the skew polynomial ring R[x; D] with coefficients in R and with respect to D, and show that J(R[x; D]) ∩ R is a nil D-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when R is commutative.     

Semigroup Rings as Weakly Factorial Domains

Let D be an integral domain, Γ be a torsion-free grading monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. We show that if G is of type (0, 0, 0,…), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD-domain and Γ is a weakly factorial GCD-semigroup. Let ? be the field of real numbers and Γ be the additive semigroup of nonnegative rational numbers. We also show that Γ is a weakly factorial GCD-semigroup, but ?[Γ] is not a weakly factorial domain.     

A Note on Semilocal Group Rings

Let R be an associative ring with identity and let J(R) denote the Jacobson radical of R. R is said to be semilocal if R/J(R) is Artinian. In this paper we give necessary and sufficient conditions for the group ring RG, where G is an abelian group, to be semilocal.     

The Linear Part of a Discontinuously Acting Euclidean Semigroup

Let ⁿ be a Euclidean space and let S be a Euclidean semigroup, i.e., a subsemigroup of the group of isometries of ⁿ. We say that a semigroup S acts discontinuously on ⁿ if the subset {s  S:sK ∩ K ≠ } is finite for any compact set K of ⁿ. The main results of this work areTheorem.If S is a Euclidean semigroup which acts discontinuously on ⁿ, then the connected component of the closure of the linear part ℓ(S) of S is a reducible group.Corollary.Let S be a Euclidean semigroup acting discontinuously on ⁿ; then the linear part ℓ(S) of S is not dense in the orthogonal group O(n).These results are the first step in the proof of the followingMargulis' Conjecture.If S is a crystallographic Euclidean semigroup, then S is a group.     

On the Multiplicative Quasi-Adequate Transversals of an Abundant Semigroup

In this article, we explore the multiplicative quasi-adequate transversals of an abundant semigroup. Let S be an abundant semigroup with multiplicative quasi-adequate transversals. Then the product of any two multiplicative quasi-adequate transversals is also a multiplicative quasi-adequate transversal. Moreover, all multiplicative quasi-adequate transversals of S form a rectangular band. Let S°and S ^? be multiplicative quasi-adequate transversals of S, and let δ°(δ^?) be the δ-relation on S°(S ^?). Then there exists a bijection ? from S°/δ°onto S ^?/δ^?. In particular, if δ°and δ^? are congruences, then the bijection ? is an isomorphism.     

On the Partially Ordered Semigroup Generated by the Class Operators I,R,H,S,P

Let I, H, S, P be the usual class operators on universal algebras. For a class K of universal algebras of the same type, let R({K}) be the class of all algebras isomorphic to a retract of a member of K and let R denote the corresponding class operator. In this paper the semigroup generated by class operators I, R, H, S, P and the corresponding partially ordered set are described. Also the standard semigroups of the above operators are determined for some varieties.     

Divisorial Linear Algebra of Normal Semigroup Rings

We investigate the minimal number of generators and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound C there exist, up to isomorphism, only finitely many divisorial ideals I such that (I)C. It follows that there exist only finitely many Cohen–Macaulay divisor classes. Moreover, we determine the minimal depth of all divisorial ideals and the behaviour of and depth in arithmetic progressions in the divisor class group.The results are generalized to more general systems of linear inequalities whose homogeneous versions define the semigroup in a not necessarily irredundant way. The ideals arising this way can also be considered as defined by the nonnegative solutions of an inhomogeneous system of linear diophantine equations.We also give a more ring-theoretic approach to the theorem on minimal number of generators of divisorial ideals: it turns out to be a special instance of a theorem on the growth of multigraded Hilbert functions.     

Irreducibility and Reduction Number of Ideals in a One Dimensional Analytically Irreducible Ring

Let R be a one dimensional analytically irreducible ring, and let I be an integral ideal of R. We study the irreducibility and the reduction number of I in relation with the corresponding semigroup ideal v(I) in v(R), where v(R) is the semigroup of values of R. It turns out that, if v(I) is irreducible, then I is irreducible, but the converse does not hold in general. We show also that the reduction number of I in R can assume any positive value less than the multiplicity of R and can be different from the reduction number of v(I).     

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.     

Rees Matrix Covers and the Translational Hull of a Locally Inverse Semigroup

For any locally inverse semigroup S, there exists a maximal dense ideal extension of S within the class LI of all locally inverse semigroups (Pastijn and Oliveira, 2006 Pastijn , F. J. , Oliveira , L. ( 2006 ). Maximal dense ideal extensions of locally inverse semigroups . Semigroup Forum 72 : 441 – 458 . [Google Scholar], Preprint). Here we realize this maximal dense ideal extension in terms of a canonically constructed quotient of a regular Rees matrix semigroup over an inverse semigroup.     

Continuous extension of a densely parameterized semigroup

Let be a dense sub-semigroup of ℝ₊, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over can be extended to a weakly continuous semigroup over ℝ₊. We obtain similar results for nonlinear, nonexpansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over ℝ₊. O.M. Shalit was partially supported by the Gutwirth Fellowship.     

The regularity of munn rings and semigroup rings

In this paper, the author studies the regularity of Munn rings and completely 0-simple semigroup rings. When either (i)R is a ring with identity andI and are infinite, or (ii)R is a strong IBN and fully Dedekind-finite ring with identity and eitherI or is finite, in Section 2, the regularity of the Munn ringM(R; I, ; P) is characterized; in Section 3, for a completely 0-simple semigroupS =M°(G; I, ; P), the regularity ofRS is characterized. Meantime, the author shows that for a locally finite monoidS and a ringR with identity,RS is a strong IBN ring if and only ifR is so;RS is fully Dedekind-finite if and only ifR is so.     

局部单位元半群分次环

设S为有限局部单位元半群,R为S—分次环.首先定义了S—分次环R在半群S上的冲积R#S*,证明了模范畴R#S*-M od与分次模范畴(S,R)-g r之间的等价性,并进一步研究了局部单位元半群分次环的分次Jacobson根及其相关的自反根的关系,得到重要关系式J(R#S*)=JS(R)#S*及Jref(R)=(J(R#S*))↓=JS(R).     

Maximal Homomorphic Group Image and Convergence of Convolution Sequences on a Semigroup

Let be a probability measure generating a locally compact semigroup S. If the convolution sequence ⁿ is tight, in particular if S is compact, S admits a closed minimal ideal K. The convergence of ⁿ is characterized in terms of convergence of a homomorphic image (~) ⁿ on a factor group of the compact group G in the Rees–Suschkewitsch decomposition of K.