弱交换富足序半群(Ⅰ)

本文将序半群上的 Ｇｒｅｅｎ’ｓ－关系推广为 Ｇｒｅｅｎ’ｓ＊一关系．给出主序（左、右）＊－理想、主序＊－滤特征描述和弱交换富足序半群的特征．用这些特征证明了一类弱交换富足序半群的结构定理：若序半群Ｓ满足 ,则Ｓ是弱交换富足序半群当且仅当Ｓ是左（右）单序半群｛（ｅ）（Ｓ）｝的半格．     

Bands of weakly r-Archimedean ordered semigroups

In this paper, some characterizations that an ordered semigroup S is a band of weakly r-archimedean ordered subsemigroups of S are given by some relations on S . We prove that an ordered semigroup S is a band of weakly r -archimedean ordered subsemigroups if and only if S is regular band of weakly r -archimedean ordered subsemigroups. Finally, we obtain that a negative ordered semigroup S is a band of weakly r-archimedean ordered subsemigroups of S if and only if S is a band of r-archimedean ordered subsemigroups of S . As an application the corresponding results on semigroups without order can be obtained by moderate modifications. August 27, 1999     

On left regular and intra-regular ordered semigroups

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.     

关于弱交换po－半群

在本文中我们引入弱交换ｐｏ－半群的概念，并研究这类半群到其Ａｒｃｈｉｍｅｄｅｓ子半群的半格分解，给出了这类半群类似于无序半群相应结果的一个刻画。作为推论，我们得到弱交换ｐｏｅ－群和无序半群的相应刻画。     

关于弱交换PO—半群中的一个问题

本文引入弱交换ｐｏ－半群的概论２，研究这类半群到Ａｒｃｈｉｍｅｄｅａｎ子半群的半格分解，得到了这半群类似于具平凡序的弱交换半群的一个特征，由此在更一般的情形下回答了Ｋｅｈａｙｏｐｕｌｕ在「１」中提出的一个问题，并作为推论得到弱交换ｐｏｅ－半群和具平凡序的弱交换半群的已知结果。     

Chain decompositions of ordered semigroups

Characterizations of ordered semigroups which can be decomposed into (natural ordered) chains of ω -simple ordered semigroups are given, where ω -simple ordered semigroups are ξ _l (ξ _t ) -simple, left (t -) simple, L _n (H _n ) -simple, l (t )-archimedean and nil-extensions of left (t -) simple ordered semigroups, respectively. As a generalization of the theory of Clifford semigroups (without orders) to ordered semigroups, ordered semigroups which are semilattices of t -simple subsemigroups are characterized.     

On the Least Property of the Semilattice Congruences on PO-Semigroups

n on po-semigroups. We study the least property of (ordered) semilattice congruences, and prove: 1. N is the least ordered semilattice congruence on pr-semigroups (cf.[1]). 2. n is the least semilattice congruence on po-semigroups. 3. N is not the least semilattice congruence on po-semigroups in general. Thus, we give a complete solution to the problem posed by N. Kehayopulu in [1].     

Nil-extensions of simple po-semigroups

In this paper, we generalize the concepts of (intra-, left, right, completely) π-regular semigroups without order to po-semigroups and discuss characterizations and relationships concerning them. Moreover, we introduce the concepts of nil-extensions of po-semigroups first, then consider properties and characterizations of po-semigroups which are nil-extensions of (left, rightt-) simple (π-regular) po-semigroups and complete semilattices of this class of po-semigroups.     

几类亚交换序半群

本文研究几类亚交换序半群的性质,并将具有单位元的亚交换序半群的一些结果扩张到这几类亚交换序半群上,使得这些结果更加细化,其中主要证明了以下定理：伪交换序半群可以分解成阿基米德序半群的半格．并且,一般来说,这种分解不是唯一的．     

On Ordered Semigroups Which are Semilattices of Simple and Regular Semigroups

We characterize the ordered semigroups which are decomposable into simple and regular components. We prove that each ordered semigroup which is both regular and intra-regular is decomposable into simple and regular semigroups, and the converse statement also holds. We also prove that an ordered semigroup S is both regular and intra-regular if and only if every bi-ideal of S is an intra-regular (resp. semisimple) subsemigroup of S. An ordered semigroup S is both regular and intra-regular if and only if the left (resp. right) ideals of S are right (resp. left) quasi-regular subsemigroups of S. We characterize the chains of simple and regular semigroups, and we prove that S is a complete semilattice of simple and regular semigroups if and only if S is a semilattice of simple and regular semigroups. While a semigroup which is both π-regular and intra-regular is a semilattice of simple and regular semigroups, this does not hold in ordered semigroups, in general.     

Operator-valued Feynman integral via conditional Feynman integrals on a function space

Let C ₀ ^r [0; t] denote the analogue of the r-dimensional Wiener space, define X _t : C ^r [0; t] → ?^2r by X _t (x) = (x(0); x(t)). In this paper, we introduce a simple formula for the conditional expectations with the conditioning function X _t . Using this formula, we evaluate the conditional analytic Feynman integral for the functional $$ \Gamma _t \left( x \right) = exp \left\{ {\int_0^t {\theta \left( {s,x\left( s \right)} \right)d\eta \left( s \right)} } \right\}\varphi \left( {x\left( t \right)} \right) x \in C^r \left[ {0,t} \right] $$ , where η is a complex Borel measure on [0, t], and θ(s, ·) and φ are the Fourier-Stieltjes transforms of the complex Borel measures on ? ^r . We then introduce an integral transform as an analytic operator-valued Feynman integral over C ^r [0, t], and evaluate the integral transform for the function Γ _t via the conditional analytic Feynman integral as a kernel.     

Weakly separative weakly commutative semigroups

A semigroup S is called a weakly commutative semigroup if, for every a,b∈S, there is a positive integer n such that (ab) ⁿ ∈Sa∩bS. A semigroup S is called archimedean if, for every a,b∈S, there are positive integers m and n such that a ⁿ ∈SbS and b ^m ∈SaS. It is known that every weakly commutative semigroup is a semilattice of weakly commutative archimedean semigroups. A semigroup S is called a weakly separative semigroup if, for every a,b∈S, the assumption a ²=ab=b ² implies a=b. In this paper we show that a weakly commutative semigroup is weakly separative if and only if its archimedean components are weakly cancellative. This result is a generalization of Theorem 4.16 of Clifford and Preston (The Algebraic Theory of Semigroups, Am. Math. Soc., Providence, 1961).     

阿基米德序半群的链的若干刻画

本文首先引入了一个序半群$S$的准素模糊理想的概念,通过序半群$S$上的一些二元关系以及它的理想的模糊根给出了该序半群是阿基米德序子半群的半格的一些刻画.进一步地借助于序半群$S$的模糊子集对该序半群是阿基米德序子半群的半格进行了刻画.尤其是通过序半群的模糊素根定理证明了序半群$S$是阿基米德序子半群的链当且仅当$S$是阿基米德序子半群的半格且$S$的所有弱完全素模糊理想关于模糊集的包含关系构成链.     

Ordered semigroups which are both right commutative and right cancellative

In this paper we prove that each right commutative, right cancellative ordered semigroup (S,.,??) can be embedded into a right cancellative ordered semigroup (T,??,?) such that (T,??) is left simple and right commutative. As a consequence, an ordered semigroup S which is both right commutative and right cancellative is embedded into an ordered semigroup T which is union of pairwise disjoint abelian groups, indexed by a left zero subsemigroup of?T.     

Completeness and basis properties of systems of exponentials in weighted spaces L p(−π, π)

We consider the system of exponentials $e(\Lambda ) = \{ e^{i\lambda _n t} \} _{n \in \mathbb{Z}} $ , where $$\lambda _n = n + \left( {\frac{{1 + \alpha }}{p} + l(\left| n \right|)} \right) sign n,$$ l(t) is a slowly varying function, and l(t) → 0, t → ∞. We obtain an estimate for the generating function of the sequence {λ_n} and, with its help, find a completeness criterion and a basis condition for the system e(Λ) in the weight spaces L ^p(?π, π). We also study some special cases of the function l(t).     

Semilattice modes I: the associated semiring

We examine idempotent, entropic algebras (modes) which have a semilattice term. We are able to show that any variety of semilattice modes has the congruence extension property and is residually small. We refine the proof of residual smallness by showing that any variety of semilattice modes of finite type is residually countable. To each variety of semilattice modes we associate a commutative semiring satisfying 1 +r=1 whose structure determines many of the properties of the variety. This semiring is used to describe subdirectly irreducible members, clones, subvariety lattices, and free spectra of varieties of semilattice modes.Presented by J. Berman.Part of this paper was written while the author was supported by a fellowship from the Alexander von Humboldt Stiftung.     

Largest subsemigroups of the full transformation monoid

In this paper we are concerned with the following question: for a semigroup S, what is the largest size of a subsemigroup T?S where T has a given property? The semigroups S that we consider are the full transformation semigroups; all mappings from a finite set to itself under composition of mappings. The subsemigroups T that we consider are of one of the following types: left zero, right zero, completely simple, or inverse. Furthermore, we find the largest size of such subsemigroups U where the least rank of an element in U is specified. Numerous examples are given.     

Über Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1)u′(t)=F(t,u) for )≤t≤∞ together with a generalized initial condition (2)u(t)=?(t) forr≤t≤0 or a generalized Nicoletti condition (3)N u=η. Here,N is a linear operator; in the case of a system ofn equations the classical Nicoletti operator is given byN u=(u ₁(t ₁),...,u _n(t _n)), with givent _i. The functionsu, F ? are Banach space valued, the functionF(t, z) is defined fort≥0 andz∈C ⁰[r,∞). The main point is that the value ofF(t, z) may depend on the values ofz(s) forr≤s≤t+σ(t), where σ(t)>0. Simple examples show that without a restriction on the magnitude of the advancement σ(t) there is neither existence nor uniqueness. Our results show that when σ(t) is properly bounded and when the solution is to satisfy a certain growth condition which depends on σ(t), then there exists exactly one solution, and it depends continuously on the given data. In the case of the Nicoletti problem (1), (3) there is convergence to the solution satisfyingu(0)=η if 0≤t _i≤T andT→0 (this holds in infinite-dimensional spaces, too). These results are true ifF satisfies a Lipschitz condition of the form $$\left| {F(t,z) - F(t,y)} \right| \leqslant h(t)\max \left\{ {\left| {z(s) - y(s)} \right|:r \leqslant s \leqslant t + \delta (t)} \right\}.$$ . In the case where (1) is a finite system andF is only continuous, an existence theory is developped based onSchauder's fixed point theorem. Again, growth conditions play an essential role here.     

Statistical extension of classical Tauberian theorems in the case of logarithmic summability

Let s: [1,∞) → ? be a locally integrable function in Lebesgue’s sense. The logarithmic (also called harmonic) mean of the function s is defined by $$\tau (t): = \frac{1} {{\log t}}\int_1^t {\frac{{s(x)}} {x}dx, t > 1,}$$ where the logarithm is to the natural base e. Besides the ordinary limit lim _x→∞ s(x), we use the notion of the so-called statistical limit of s at ∞, in notation: st-lim _x→∞ s(x) = l, by which we mean that for every ? > 0, $$\mathop {\lim }\limits_{b \to \infty } \frac{1} {b}\left| {\left\{ {x \in (1,b):\left| {s(x) - \ell } \right| > \varepsilon } \right\}} \right| = 0.$$ We also use the ordinary limit limt→∞ τ (t) as well as the statistical limit st-limt→∞ τ (t). We will prove the following Tauberian theorem: Suppose that the real-valued function s is slowly decreasing or the complex-valued s is slowly oscillating. If the statistical limit st-limtt→∞ τ (t) = l exists, then the ordinary limit limx→∞ s (x) = l also exists.