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1.
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.  相似文献   

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3.
《代数通讯》2013,41(6):2461-2479
Superabundant semigroups are generalizations of completely regular semigroups written the class of abundant semigroups. It has been shown by Fountain that an abundant semigroup is superabundant if and only if it is a semilattice of completely J *-simple semigroups. Reilly and Petrich called a semigroup S cryptic if the Green's relation H is a congruence on S. In this paper, we call a superabundant semigroup S a regular crypto semigroup if H * is a congruence on S such that S/H * is a regular band. It will be proved that a superabundant semigroup S is a regular crypto semigroup if and only if S is a refined semilattice of completely J *-simple semigroups. Thus, regular crypto semigroups are generalization of the cryptic semigroups as well as abundant semigroups.  相似文献   

4.
It has been proved by Tôru Saitô that a semigroup S is a semilattice of left simple semigroups, that is, it is decomposable into left simple semigroups, if and only if the set of left ideals of S is a semilattice under the multiplication of subsets, and that this is equivalent to say that S is left regular and every left ideal of S is two-sided. Besides, S. Lajos has proved that a semigroup S is left regular and the left ideals of S are two-sided if and only if for any two left ideals L 1, L 2 of S, we have L 1L 2 = L 1 L 2. The present paper generalizes these results in case of ordered semigroups. Some additional information concerning the semigroups (without order) are also obtained.  相似文献   

5.
证明了H~#-富足半群S是正规密码H~#-富足半群当且仅当它是完全J~#-单半群的强半格.该结果也是正规密码超富足半群和正规密码群并半群分别在超富足半群和完全正则半群上的相应结构定理的推广.  相似文献   

6.
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.  相似文献   

7.
证明了ο-超富足半群S是正规密码ο-超富足半群当且仅当它是完全Jο-单半群的强半格.该结果也是正规密码超富足半群和正规密码群并半群分别在超富足半群和完全正则半群上的相应结构定理的推广。  相似文献   

8.
A cryptogroup is a completely regular semigroup S on which Green’s relation $\mathcal{H}$ is a congruence. For a,xS, x is an associate of a if a=axa. A subgroup G of S is an associate subgroup of S if it contains precisely one associate of each element of S. Further, S is a regular (respectively normal) cryptogroup if $S/\mathcal{H}$ is a regular (respectively normal) band. We provide a construction of a general (respectively regular or normal) cryptogroup in terms of groups and functions. On this model of S, we find several conditions equivalent to S containing an associate subgroup G. We characterize several varieties of completely regular semigroups, provided with the unary operation s?s ?, where s ? is the associate of s in G. They include completely regular semigroups, (regular, normal) cryptogroups, completely simple semigroups, and their monoid and/or overabelian members.  相似文献   

9.
In this paper we investigate how the combinatorial property finite derivation type (FDT) is preserved in a semilattice of semigroups. We prove that if $S= \mathcal{S}[Y,S_{\alpha}]$ is a semilattice of semigroups such that Y is finite and each S ?? (????Y) has FDT, then S has FDT. As a consequence we can show that a strong semilattice of semigroups $\mathcal{S}[Y,S_{\alpha},\lambda_{\alpha,\beta}]$ has FDT if and only if Y is finite and every semigroup S ?? (????Y) has FDT.  相似文献   

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We show that if Y is a subsemilattice of a finite semilattice indecomposable semigroup S then \({|Y|\leq 2\left\lfloor \frac{|S|-1}{4}\right\rfloor+1}\). We also characterize finite semilattice indecomposable semigroups S which contain a subsemilattice Y with \({|S|=4k+1}\) and \({|Y|=2\left\lfloor \frac{|S|-1}{4} \right\rfloor+1=2k+1}\). They are special inverse semigroups. Our investigation is based on our new result proved in this paper which characterizes finite semilattice indecomposable semigroups with a zero by using only the properties of its semigroup algebra.  相似文献   

12.
A semigroup S is called η-simple if S has no semilattice congruences except S×S. Tamura in (Semigroup Forum 24:77–82, 1982) studied η-simple semigroups with a unique idempotent. In the present paper we consider a more general situation, that is, we investigate η-simple semigroups (without zero) with a least idempotent. Moreover, we study η ?-simple semigroups with zero which contain a least non-zero idempotent.  相似文献   

13.
On any regular semigroup S, the greatest idempotent pure congruence τ the greatest idempotent separating congruence μ and the least band congruence β are used to give the S-classification of regular semigroups as follows. These congruences generate a sublattice Λ of the congruence lattice C(S) of S. We consider the triples (Λ,K,T), where K and T are the restrictions of the K- and T-relations on C(S) to Λ. Such triples are characterized abstractly and form the objects of a category S whose morphisms are surjective K- and T-preserving homomorphisms subject to a mild condition. The class of regular semigroups is made into a category S whose morphisms are fairly restricted homomorphisms. The main result of the paper is the existence of a representative functor from S to S. The effect of the S-classification on Reilly semigroups and cryptogroups is discussed briefly.  相似文献   

14.
ABSTRACT

The investigation of regular F-abundant semigroups is initiated. In fact, F-abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F-abundant semigroups is obtained. In addition, we also prove that a regular F-abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F-inverse semigroups.  相似文献   

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Ortho-u-monoids     
In this paper, we study the class of ortho-u-monoids which are generalized orthogroups within the class of E(S)-semiabundant semigroups. After introducing the concept of (∼)-Green’s relations, and obtaining some important properties of (∼)-Green’s relations and super E(S)-semiabundant semigroups, we have given the semilattice decomposition of ortho-u-monoids and a structure theorem for regular ortho-u-monoids. The main techniques that we used in the study are the (∼)-Green’s relations, and the semi-spined product of semigroups.  相似文献   

17.
Wlpp semigroups are generalizations of lpp semigroups and regular semi-groups. In this paper, we consider some kinds of wlpp semigroups, namely right-e wlpp semigroups. It is proved that such a semigroup S , if and only if S is the strong semilattice of L-right cancellative planks;also if and only if S is a spined product of a right-e wlpp semigroup and a left normal band.  相似文献   

18.
LetS be a semigroup;S is said to bepermutable if, for some integern, every product ofn elements ofS can be re-ordered. We prove that every normal extension of a semilattice by an inverse permutable semigroupsis permutable. Also, some properties of permutable groups are extended to inverse semigroups.  相似文献   

19.
《代数通讯》2013,41(8):2929-2948
Abstract

A semigroup S is called E-inversive if for every a ∈ S there is an x ∈ S such that ax is idempotent. The purpose of this paper is the investigation of E-inversive semigroups and semigroups whose idempotents form a subsemigroup. Basic properties are analysed and, in particular, semigroups whose idempotents form a semilattice or a rectangular band are considered. To provide examples and characterizations, the construction methods of generalized Rees matrix semigroups and semidirect products are employed.  相似文献   

20.
推广了半群的强半格分解的定义,得到了半群的拟强半格分解,并证明了完全正则半群为群 的正则(或右拟正规)带当且仅当它是完全单半群的拟强半格(且 )).  相似文献   

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