Semilattice Decompositions Research Articles

A Characterization of $LR$-regular Bands

In this paper, some characterizations of LR-regular bands, WLR-regular bands and some subclasses of them are given. Furthermore, we investigate the refined semilattice decompositions of them. Keywords: LR-regular band; refined semilattice; LR-normal band MR(2010) Subject Classification: 20M17; 20M10 / CLC number: O152.7 Document code: A Article ID: 1000-0917(2015)03-0379-08 pair α, β ∈ Y such that α ≥ β ,l etD(α, β) be a set of index and {Sd(α,β) : d(α, β) ∈ D(α, β)} be a congruence partition of Sβ (i.e., the relation ρ on Sβ defined by (b1 ,b 2) ∈ ρ if and only if b1 ,b 2 ∈ Sd(α,β) for some d(α, β) ∈ D(α, β) is a congruence on Sβ). And for any α, β, γ ∈ Y with α ≥ β ≥ γ, the partition {Sd(α,γ) : d(α, γ) ∈ D(α, γ)} is dense in the partition {Sd(β,γ) : d(β, γ) ∈ D(β, γ)}, i.e., for any d(β, γ) ∈ D(β, γ), there exists D � (α, γ) ⊆ D(α, γ) such that

Strong quasi-ideals and complete semilattice decompositions of an ordered semigroup

There are quasi-ideals of an ordered semigroup which cannot be expressed as an intersection of a left ideal and a right ideal. A quasi-ideal with this intersection property is called strong quasi-ideal. We show that strong quasi-simplicity is equivalent to t-simplicity of an ordered semigroup; and hence it turns out to be the case that the ordered semigroups which are complete semilattices (chains) of t-simple subsemigroups can be characterized by their strong quasi-ideals. An ordered semigroup S is complete semilattice (chain) of t-simple subsemigroups if and only if every strong quasi-ideal of S is a completely semiprime (prime) ideal. Also we introduce and characterize the minimal strong quasi-ideals.

Commutative orders revisited

This article studies commutative orders, that is, commutative semigroups having a semigroup of quotients. In a commutative order \(S\), the square-cancellable elements \(\mathcal {S}(S)\) constitute a well-behaved separable subsemigroup. Indeed, \(\mathcal {S}(S)\) is also an order and has a maximum semigroup of quotients \(R\), which is Clifford. We present a new characterisation of commutative orders in terms of semilattice decompositions of \(\mathcal {S}(S)\) and families of ideals of \(S\). We investigate the role of tensor products in constructing quotients, and show that all semigroups of quotients of \(S\) are homomorphic images of the tensor product \(R\otimes _{\mathcal {S}(S)} S\). By introducing the notions of generalised order and semigroup of generalised quotients, we show that if \(S\) has a semigroup of generalised quotients, then it has a greatest one. For this we determine those semilattice congruences on \(\mathcal {S}(S)\) that are restrictions of congruences on \(S\).

Band and semilattice decompositions of Xk-simple ordered semigroups

In this paper, we define various types of k-regularity of ordered semigroups and using generalizations of some Green?s relations and its properties, are described the structure of band and complete semilattice of Lk-simple, Ik-simple, Bk-simple, Hk-simple ordered semigroups.

SEMILATTICES OF SUBDIMONOIDS

We present some congruence on the dimonoid with an idempotent operation and use it to obtain semilattice decompositions of an idempotent dimonoid. Also we give necessary and sufficient conditions under which an arbitrary dimonoid is a semilattice of archimedean subdimonoids.

Completely Regular Semigroups with Generalized Strong Semilattice Decompositions

The concept of ρG-strong semilattice of semigroups is introduced. By using this concept, we study Green's relation ℋ on a completely regular semigroup S. Necessary and sufficient conditions for S/ℋ to be a regular band or a right quasi-normal band are obtained. Important results of Petrich and Reilly on regular cryptic semigroups are generalized and enriched. In particular, characterization theorems of regular cryptogroups and normal cryptogroups are obtained.

Semigroups of Left Quotients—The Layered Approach#

A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be expressed as a ♯ b where a, b ∈ S and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. Here a ♯ denotes the inverse of a in a subgroup of Q. We say that a left order S is straight in Q if in the above definition we can insist that a ℛ b in Q. A complete characterisation of straight left orders in terms of embeddable ∗-pairs is available. In this article we adopt a different approach, based on partial order decompositions of semigroups. Such decompositions include semilattice decompositions and decompositions of a semigroup into principal factors or principal ∗-factors. We determine when a semigroup that can be decomposed into straight left orders is itself a straight left order. This technique gives a unified approach to obtaining many of the early results on characterisations of straight left orders.

The construction of orthodox super rpp semigroups

We define orthodox super rpp semigroups and study their semilattice decompositions. Standard representation theorem of orthodox super rpp semigroups whose subband of idempotents is in the varieties of bands described by an identity with at most three variables are obtained.

Semilattice Decompositions of Semigroups Revisited

Two relations introduced by M. S. Putcha and T. Tamura, denoted by A! and , play a crucial role in semilattice decompositions of semigroups. General properties of the graphs that correspond to these relations were studied by M. S. Putcha in [14], 1974, and the structure of semigroups in which the minimal paths in the graph corresponding to A! are bounded was described by the Þrst two authors in [7], 1996. In this paper this will be done for the relation . We consider the same problem for the left-hand analogue of this relation and for two radicals of the Green’s J -relation. The obtained results generalize various results given by M. S. Putcha, T. Tamura, L. N. Shevrin and the Þrst two authors of this paper.

Remarks on varieties of involution bands

In the present paper, we study varieties consisting of bands (idempotent semigroups) endowed with an involutorial antiautomorphism * as a fundamental operation. Our principal aim is here to provide an insight to some classes of these varieties from the structural point of view, especially in terms of semilattice decompositions, subdirect products and ideal extensions. In the course of such considerations, we shall extend the result of C. L. Adair [1], who described the lattice of all varieties of bands with a regular involution (i.e. with the identity x = xx * x) We depict a broader lattice of involution band varieties, which incorporates Adair’s lattice.

On weak commutativity of po-semigroups and their semilattice decompositions

In this paper, concepts of left (right) weakly commutative po-semigroups and r (l or t)-archimedean subsemigroups of a po-semigroup are introduced. Six relations τ, σ, η, ρ, μ, ξ on a po-semigroup are defined. By using them, filters and radicals, fourteen necessary and sufficient conditions in order that a po-semigroup is a semilattice of archimedean subsemigroups are given. The facts that a left weakly commutative (right weakly commutative or weakly commutative) po-semigroup is a semilattice of r (l or t) -archimedean subsemigroups are proved and seven characterizations of these po-semigroups are obtained respectively.

Semilattice decompositions of semigroups

Semilattice decompositions of semigroups

The structure of left C-rpp semigroups

This paper studies the class of left Clifford-rpp semigroups and investigates the structure of their semi-spined products and semilattice decompositions. These semigroups are generalizations of left Clifford semigroups and Clifford-rpp semigroups. We also discuss some special cases such as when a semilattice decomposition becomes a strong semilattice decomposition and a semi-spined product becomes a spined product.

STRUCTURE AND CHARACTERISTICS OF LEFT CLIFFORD SEMIGROUPS

As generalization of Clifford semigroups, left Clifford semigroups are defined and ξ-products for such semigroups and their semilattice decompositions are studied. In particular, considering how a semilattice decomposition becomes a strong semilattice decomposition and ξ-product becomes spined product, some structure theorems and characteristics for this class of semigroups are obtained.

On a class of Dubreil-Jacotin regular semigroups and a construction of Yamada

SynopsisIn the publication [2] we obtained some structure theorems for certain Dubreil-Jacotin regular semigroups. A crucial observation in the course of investigating these types of ordered regular semigroups was that the (ordered) band of idempotents was normal. This is characteristic of a class of semigroups studied by Yamada [5] and called generalised inverse semigroups. Here we specialise a construction of Yamada to obtain a structure theorem that complements those in [2], The important feature of the present approach is the part played by the greatest elements that exist in each of the components in the semilattice decompositions involved.

Quasi‐orders, Generalized Archimedeaness and Semilattice Decompositions

Quasi‐orders, Generalized Archimedeaness and Semilattice Decompositions

Semilattice decompositions of semigroups

The purpose of this paper is to develop a general theory of semilattice decompositions of semigroups from the point of view of obtaining theorems of the type: A semigroup S has propertyD if and only if S is a semilattice of semigroups having property β. As such we are able to extend the theories of Clifford [3], Andersen [1], Croisot [5], Tamura and Kimura [14], Petrich [9], Chrislock [2], Tamura and Shafer [15], Iyengar [7] and Weissglass and the author [10]. The root of our whole theory is Tamura's semilattice decomposition theorem [12, 13]. Of this, we give a new proof.

The maximal semilattice decomposition of a semigroup

A semigroup is a non-empty set on which an associative multiplication is defined. A semilattice is a commutative semigroup each of whose elements is idempotent. The maximal semilattice homomorphic image of an arbitrary semigroup S is the semilattice Y such that every semilattice homomorphic image of S is also a homomorphic image of Y. The maximal semilattice decomposition of S is the decomposition of S into equivalence classes which are complete inverse images of members of Y. We identify these classes with members of Y. YAMADA [20] and CLIFFORD [3] have characterized such a decomposition for an arbitrary semigroup, TnlERRIN [18] for a groupoid, TAMURA and KIMURA [15] for a commutative semigroup, and MCLEAN [11] for a band. We give several characterizations of the maximal semilattice decomposition for arbitrary semigroups and others for some special classes of semigroups. Further, we consider some properties of members of such a decomposition. In section 2 we give a characterization of all semilattice decompositions of S in terms of prime ideals of S (2.3). In section 3 we show that a member N~ (containing x~S) of the maximal semilattice decomposition Y of S is the set of all elements y of S such that the smallest face of S containing x is also the smallest face of S containing y (3.2). Moreover, we show that no ideal of any N~ contains a proper prime ideal (3.4). These results are of fundamental importance in the remainder of the paper. We prove, among other things, that N x is the largest subsemigroup of S containing x and containing no proper prime ideals (3.10). In section 4 we establish several properties of each N, in terms of properties of elements of S and in terms of (several kinds of) ideals of S. Here we partly make use of the decomposition theory of CROISOT [6] and CLIFFORD [1], [2]. In section 5 we perform a similar analysis as in section 4 under the additional hypothesis that Y be linearly ordered. Finally, in section 6 we give explicit expressions for the smallest face of S containing an element x of S for some classes of semigroups. In particular we do this for a class of semigroups considered by THIERRIN [19] (6.1), a class of semigroups which we call weakly commutative (6.5), and a class of periodic semigroups (6.7 and 6.9).

The maximal semilattice decomposition of a semigroup

A semigroup is a non-empty set on which an associative multiplication is defined. A semilattice is a commutative semigroup each of whose elements is idempotent. The maximal semilattice homomorphic image of an arbitrary semigroup S is the semilattice Y such that every semilattice homomorphic image of S is also a homomorphic image of Y. The maximal semilattice decomposition of S is the decomposition of S into equivalence classes which are complete inverse images of members of Y. We identify these classes with members of Y. YAMADA [20] and CLIFFORD [3] have characterized such a decomposition for an arbitrary semigroup, TnlERRIN [18] for a groupoid, TAMURA and KIMURA [15] for a commutative semigroup, and MCLEAN [11] for a band. We give several characterizations of the maximal semilattice decomposition for arbitrary semigroups and others for some special classes of semigroups. Further, we consider some properties of members of such a decomposition. In section 2 we give a characterization of all semilattice decompositions of S in terms of prime ideals of S (2.3). In section 3 we show that a member N~ (containing x~S) of the maximal semilattice decomposition Y of S is the set of all elements y of S such that the smallest face of S containing x is also the smallest face of S containing y (3.2). Moreover, we show that no ideal of any N~ contains a proper prime ideal (3.4). These results are of fundamental importance in the remainder of the paper. We prove, among other things, that N x is the largest subsemigroup of S containing x and containing no proper prime ideals (3.10). In section 4 we establish several properties of each N, in terms of properties of elements of S and in terms of (several kinds of) ideals of S. Here we partly make use of the decomposition theory of CROISOT [6] and CLIFFORD [1], [2]. In section 5 we perform a similar analysis as in section 4 under the additional hypothesis that Y be linearly ordered. Finally, in section 6 we give explicit expressions for the smallest face of S containing an element x of S for some classes of semigroups. In particular we do this for a class of semigroups considered by THIERRIN [19] (6.1), a class of semigroups which we call weakly commutative (6.5), and a class of periodic semigroups (6.7 and 6.9).