On Irreducibles in the Polynomial Semigroup

We consider the asymptotic behavior of the number of irreducible polynomials over a finite field in arithmetic progressions.     

On Generalized Semigroup Near-Rings

Using Meldrum and van der Walt's scheme we successfully define (generalized) semigroup near-rings which are the extensions of their ring counterpart. Some standard semigroup ring results are extended. We define contracted objects for (generalized) semigroup near-rings and show (generalized) matrix near-rings are just a special case as in rings. This theory encompasses (generalized) matrix, group, and (generalized) polynomial near-rings.     

On the Products of Three Theta Functions

In this paper, we re-examine several beautiful identities of Ramanujan from the perspective of the products of three theta functions.     

On the Semigroup Algebra of Binary Relations

The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B _n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB _n contains an ideal K _n of dimension (2 ⁿ  ? 1)², and we construct an explicit isomorphism of K _n with the matrix algebra M _{2 ⁿ ?1}(F).     

关于蕴涵BCK—代数的伴随半群

我们证明了蕴涵BCK-代数的伴随半群是一个上半格;具有条件（s)的蕴涵BCK-代数的伴随半群是一个广义布尔代数。更进一步证明了有界蕴涵BCK-代数的伴随半群是一个布尔代数。     

On Semigroup Presentations and Efficiency

The purpose of this paper is twofold. First we determine some forms of the relations in a finite semigroup presentation with zero deficiency which does or does not define a group. Moreover, we conclude that a finite Rees matrix semigroup M [G; I, Λ; P] is efficient when G is efficient and the index sets I, Λ are finite.     

半群SOP_n的平方幂等元秩

设SOP_n是M上的奇异方向保序交换半群.证明了半群SOP_n是由秩为n-1的平方幂等元生成的,且它的平方幂等元秩为n.     

关于有限群极大子群的强θ-完备

本文定义了有限群极大子群的强θ-完备这一概念.通过研究极大子群的强θ-完备对群结构的影响,给出了有限群可解性,超可解性,幂零性的一些新刻画,所得结果改进了以往的定理.     

关于有限群极大子群的强θ-完备

本文定义了有限群极大子群的强θ-完备这一概念．通过研究极大子群的强θ-完备对群结构的影响,给出了有限群可解性,超可解性,幂零性的一些新刻画,所得结果改进了以往的定理．     

On Linear Independence of Theta Values

We prove a linear independence result for the values of theta series and Tschakaloff functions T_q(z) with different values of q.The first named author was supported in part by Grant-in-Aid for Scientific Research (No. 13640007), The Ministry of Education, Science, Sports, Culture of Japan.     

On the Trace of a Perturbed Operator Semigroup

Mathematical Notes -     

On Linear Independence of Theta Values

We prove a linear independence result for the values of theta series and Tschakaloff functions T_q(z) with different values of q.     

On the Ultrafilter Semigroup of a Topological Group

The ultrafilter semigroup Ult(G) of a topological group G consists of all nonprincipal ultrafilters on G converging to the identity and is a closed subsemigroup in the Stone-Cech compactification βG_d of G as a discrete semigroup. We show that it is consistent with ZFC that for every countable nondiscrete topological group G, Ult(G) can be partitioned into closed right ideals each of which admits a continuous homomorphism onto the Bohr compactification of the integers.     

On the Finite Base Property for Semigroup Varieties

No algorithm determines from an arbitrary recursive system of semigroup identities whether the variety defined by this system is finitely based.     

Radical Semigroup Rings and the Thue--Morse Semigroup

Let R be an associative ring with unit, let S be a semigroup with zero, and let RS be a contracted semigroup ring. It is proved that if RS is radical in the sense of Jacobson and if the element 1 has infinite additive order, then S is a locally finite nilsemigroup. Further, for any semigroup S, there is a semigroup T S such that the ring RT is radical in the Brown--McCoy sense. Let S be the semigroup of subwords of the sequence abbabaabbaababbab..., and let F be the two-element field. Then the ring FS is radical in the Brown--McCoy sense and semisimple in the Jacobson sense.     

半群ASn的秩

设A_n和Sing_n分别是X_n上的交错群和奇异变换半群.考虑变换半群AS_n=A_n∪Sing_n,证明了变换半群AS_n的秩为3.     

On the Index of Nilpotency of Semigroup Graded Rings

noindent We find the index of nilpotency of a strong supplementary semilattice sum of rings, R=tdspsum _{α∈ Y} R _α , where Y is a semilattice, when each R _α has index of nilpotency≤ k . Then we find the index of nilpotency of R when it is graded over a rectangular band Y and each R _α has index of nilpotency≤ k . These results are generalized to normal band graded rings. Further, we find sufficient conditions for a ring graded by a semilattice of nilpotent semigroups to have bounded index of nilpotency. We also show by examples that these conditions are necessary in some cases.