RESEARCH ANNOUNCEMENTS Block Separating Congruence Lattices . of .E-bisimple Semigroups

In 1966, Reilly[1] Characterized bisimple ω-semigroups as Bruck-Reilly extensions of Groups. Later, Munn and Reilly[2] proved that a congruence on a bisimple ω-semigroup is a group congruence or is idempotent-separating. Many author investigated the congruences on Bruck-Reilly extensions of some semigroups for about twenty years. In this paper, we study the block-separating     

Regular Bisimple ω2-semigroups

The regular semigroups S with an idempotent set Es = {e0,e1,…,en,…} such that e0 ＞ e1 ＞…＞ en ＞… is called a regular ω-semigroup. In [5] Reilly determined the structure of a regular bisimple ω-semigroup as BR(G,θ),which is the classical Bruck-Reilly extension of a group G.     

On U-ample ω-semigroups

The investigation of U-ample ω-semigroups is initiated. After obtaining some properties of such semigroups, a structure of U-ample ω-semigroups is established. It is proved that a semigroup is a U-ample ω-semigroup if and only if it can be expressed by WBR（T, 0）, namely, the weakly Bruck-Reilly extensions of a monoid T. This result not only extends and amplifies the structure theorem of bisimple inverse ω-semigroups given by N. R. Reilly, but also generalizes the structure theorem of ,-bisimple type A ω-semigroups given by U. Asibong-Ibe in 1985.     

含幺元的半格和群的Zappa-Sz\''{e}p积上的同余

Let P = E G be a Zappa-Szp product of a semilattice E with an identity and a group G. In this paper, we first introduce the concept of congruence pairs for P , and then prove that every congruence on P can be described by such a congruence pair. In fact the congruence lattice on P is lattice-isomorphic to the set of all congruence pairs for P . Finally,we characterize group congruences on P .     

Congruence Pairs of Decomposable MS-Algebras*

In this paper, the authors first introduce the concept of congruence pairs on the class of decomposable MS-algebras generalizing that for principal MS-algebras (see [13]). They show that every congruence relation θ on a decomposable MS-algebra L can be uniquely determined by a congruence pair (θ₁, θ₂), where θ₁ is a congruence on the de Morgan subalgebra L^?? of L and θ₂ is a lattice congruence on the sublattice D(L) of L. They obtain certain congruence pairs of a decomposable MS-algebra L via central elements of L. Moreover, they characterize the permutability of congruences and the strong extensions of decomposable MS-algebras in terms of congruence pairs.     

关于具有$Q$-逆断面的正则半群的同余格

In this paper, a complete congruence on the congruence lattice of regular semigroups with Q-inverse transversals is analysed. The classes of this complete congruence which are intervals are discussed and their least and greatest elements are presented clearly.     

The Isomorphism Theorem of Regular Bisimple ω~2-semigroups

In this paper we study the isomorphisms of two regular bisimple ω2-semigroups and obtain a criterion for two such semigroups to be isomorphic.     

The Isomorphism Theorem of Regular Bisimple ω2-semigroups

In this paper we study the isomorphisms of two regular bisimple ω2- semigroups and obtain a criterion for two such semigroups to be isomorphic.     

Modulo computably enumerable degrees by cupping partners

Cupping partners of an element in an upper semilattice with a greatest element 1 are those joining the element to 1. We define a congruence relation on such an upper semilattice by considering the elements having the same cupping partners as equivalent. It is interesting that this congruence relation induces a non-dense quotient structure of computably enumerable Turing degrees. Another main interesting phenomenon in this article is that on the computably enumerable degrees, this relation is different from that modulo the noncuppable ideal, though they define a same equivalent class for the computable Turing degree.     

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

An n × n ω-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.