A Construction for P-Regular Semigroups

A regular semigroup S with a special involution *, i.e., a unary operation on S satisfyingis called a regular *-semigroup[1]. It has been shown by Yamada[2] that a regular semigroup S isa regular *-semigroup if and only if it has a P-system, that is to say, there is a subset P of E(S)such that(2) As a generalization of regular semigroup and orthodox semigroup,Yamada[3] defined P-regular semigroup. Let S be a regularsemigroup. A subset P of E(S) is called a C-set in S if(c.2) (1) In this…     

A construction of weakly inverse semigroups

Let S° be an inverse semigroup with semilattice biordered set E° of idempotents and E a weakly inverse biordered set with a subsemilattice Ep = { e ∈ E ｜ arbieary f ∈ E, S（f , e） loheain in w（e）} isomorphic to E° by θ：Ep→E°. In this paper, it is proved that if arbieary f, g ∈E, f ←→ g→→ f°θD^s° g°θand there exists a mapping φ from Ep into the symmetric weakly inverse semigroup P J（E∪ S°） satisfying six appropriate conditions, then a weakly inverse semigroup ∑ can be constructed in P J（S°）, called the weakly inverse hull of a weakly inverse system （S°, E, θ, φ） with I（∑） ≌ S°, E（∑） ∽- E. Conversely, every weakly inverse semigroup can be constructed in this way. Furthermore, a sufficient and necessary condition for two weakly inverse hulls to be isomorphic is also given.     

The K°-relation on the Congruence Lattice of a Regular Semigroup with an Inverse Transversal

Let S be a regular semigroup with an inverse transversal S° and C(S) the congruence lattice of S. A relation K° on C(S) is introduced as follows: if ρ, θ∈ C(S), then we say that ρ and θ are K°-related if Ker ρ° = Ker θ° , where ρ° = ρ|S°. Expressions for the least and the greatest congruences in the same K°-class as ρ are provided. A number of equivalent conditions for K° being a congruence are given.     

Guarded and Banded Semigroups

The variety of guarded semigroups consists of all (S,·, ˉ) where (S,·) is a semigroup and x ↦ \overline{x} is a unary operation subject to four additional equations relating it to multiplication. The semigroup Pfn(X) of all partial transformations on X is a guarded semigroup if x \overline{f} = x when xf is defined and is undefined otherwise. Every guarded semigroup is a subalgebra of Pfn(X) for some X. A covering theorem of McAlister type is obtained. Free guarded semigroups are constructed paralleling Scheiblich's construction of free inverse semigroups. The variety of banded semigroups has the same signature but different equations. There is a canonical forgetful functor from guarded semigroups to banded semigroups. A semigroup underlies a banded semigroup if and only if it is a split strong semilattice of right zero semigroups. Each banded semigroup S contains a canonical subsemilattice g_⋆(S). For any given semilattice L, a construction to synthesize the general banded semigroup S with g_⋆ ≅ L is obtained.     

Limit Behaviors of Composition Convergent Sequences of Probability Measures on Compact L-X Semigroups

Let S be a compact second countable Hausdorff topological semigroup. Denote byP(S) the set of all regular probability measures defined on S, and by E(S) the setof all idempotents in S. S is called a L-X semigroup if ex =xe for every e∈E(S)and x∈S. In this paper, the limit behaviors of composition convergent sequences ofprobability measures defined on compact L-X semigroups are studied. Since compact     

RESEARCH ANNOUNCEMENTS On Permutable Band

A semigroup S is called band if every element of S is idempotent.A semigroup Sis called permutable if ρσ=σρ for every two congruences ρ and σ on S.In thispaper we prove that if S is a permutable band,then |S|≤13. Let S be a semigroup.The lattice of congruences on S is denoted by C(S).LetP∈C(S),AS,A is called p-saturated,if A=Uap.     

Inverse Functions of Number-Theoretic Functions (I)

If gf(x) =x for every x, then g is called a left inverse function of f and f is a right inverse function of g. If f is both left and right inverse function of g, then f and g are said to be mutually inverse to each other. We show that (§ 1) the following results hold. A function f has a left inverse if and only if f is univalent, a function g has a right inverse if and only if g is exhaustive, i. e., g takes every (natural) number as values. Hence f has both left and right inverse if and only if f is both univalent and exhaustive, i. e., f is a permutation on the domain of natural numbers. Let g_1 and g_2 be two left inverse functions of the function f. If for every left inverse g of f, we have $g_1(x) \leq g(x) \leq g_2(x)$, then g_1(x) is called the weak, and g_2(x) is the strong, left inverse function of f. Similarly we define the weak and the strong right inverse functions. We show that(§ 2) every strict increasing function f must possess weak and strong left inverse functions, and all of its left inverse functions must be exhaustive slow increasing (a function g(x) is slow increasing if and only if g(Sx) —Sg(x) =0, here s denotes the successor function). On the other hand, every exhaustive function g must possess weak and strong right inverse functions, and all of its right inverse functions must strict increasing. We show also that (§ 3): If f_1(x) and f_2(x) both take g(x) as their strong (weak) left inverse, then f_1(x)=f_2(x)(f_1(Sx)=f_2(Sx)). If g_1(x) and g_2(x) both take f(x) as their strong or weak right inverse, then g_1(x)=g_2(x). From these results we see that we may find a function from its strong (weak) left or right inverse function. Let there be f(c) \leq x 相似文献   

On generalized inverse transversals

Let S be a regular semigroup, S° an inverse subsemigroup of S.S° is called a generalized inverse transversal of S, if V（x）∩S°≠Ф. In this paper, some properties of this kind of semigroups are discussed. In particular, a construction theorem is obtained which contains some recent results in the literature as its special cases.     

右型A半群的F-覆盖

A right adequate semigroup of type F is defined as a right adequate semigroup which is an F-rpp semigroup. A right adequate semigroup T of type F is called an F-cover for a right type-A semigroup S if S is the image of T under an L＊-homomorphism. In this paper, we will prove that any right type-A monoid has F-covers and then establish the structure of F-covers for a given right type-A monoid. Our results extend and enrich the related results for inverse semigroups.     

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert <Emphasis Type="Italic">C</Emphasis>*-module <Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">A</Emphasis>)

We characterize A-linear symmetric and contraction module operator semigroup{Tt}t∈R+L(l2(A)),where A is a finite-dimensional C-algebra,and L(l2(A))is the C-algebra of all adjointable module maps on l2(A).Next,we introduce the concept of operator-valued quadratic forms,and give a one to one correspondence between the set of non-positive definite self-adjoint regular module operators on l2(A)and the set of non-negative densely defined A-valued quadratic forms.In the end,we obtain that a real and strongly continuous symmetric semigroup{Tt}t∈R+L(l2(A))being Markovian if and only if the associated closed densely defined A-valued quadratic form is a Dirichlet form.