scholarly journals Additively orthodox semirings with special transversals

2021 ◽  
Vol 7 (3) ◽  
pp. 4153-4167
Author(s):  
Kaiqing Huang ◽  
◽  
Yizhi Chen ◽  
Miaomiao Ren ◽  
◽  
...  
Keyword(s):  
Orthodox Semigroup ◽  
Completely Regular ◽  
Additive Reduct

<abstract><p>A semiring $ (S, +, \cdot) $ is called additively orthodox semiring if its additive reduct $ (S, +) $ is a orthodox semigroup. In this paper, by introducing some special semiring transversals as the tools, the constructions of additively orthodox semirings with a skew-ring transversal or with a generalized Clifford semiring transversal are established. Meanwhile, it is shown that an additively orthodox semiring with a generalized Clifford semiring transversal is a b-lattice of additively orthodox semirings with skew-ring transversals. Consequently, the corresponding results of Clifford semirings and generalized Clifford semirings in reference (M. K. Sen, S. K. Maity, K. P. Shum, Clifford semirings and generalized Clifford semirings, Taiwan. J. Math., 9 (2005), 433–444) and completely regular semirings in reference (S. K. Maity, M. K. Sen, K. P. Shum, On completely regular semirings, Bull. Cal. Math. Soc., 98 (2006), 319–328) are extended and strengthened.</p></abstract>

EXTENSIONS OF REGULAR ORTHOGROUPS BY INVERSE SEMIGROUPS

1995 ◽  
Vol 05 (03) ◽  
pp. 317-342 ◽  
Author(s):  
BERND BILLHARDT
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
Semidirect Product ◽  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Normal Extension ◽  
Group Congruence ◽  
Completely Regular ◽  
Orthodox Semigroups

Let V be a variety of regular orthogroups, i.e. completely regular orthodox semigroups whose band of idempotents is regular. Let S be an orthodox semigroup which is a (normal) extension of an orthogroup K from V by an inverse semigroup G, that is, there is a congruence ρ on S such that the semigroup ker ρ of all idempotent related elements of S is isomorphic to K and S/ρ≅G. It is shown that S can be embedded into an orthodox subsemigroup T of a double semidirect product A**G where A belongs to V. Moreover T itself can be chosen to be an extension of a member from V by G. If in addition ρ is a group congruence we obtain a recent result due to M.B. Szendrei [16] which says that each orthodox semigroup which is an extension of a regular orthogroup K by a group G can be embedded into a semidirect product of an orthogroup K′ by G where K′ belongs to the variety of orthogroups generated by K.

scholarly journals Bifree objects in e-varieties of strict orthodox semigroups and the lattice of strict orthodox *-semigroup varieties

1993 ◽  
Vol 35 (1) ◽  
pp. 25-37 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Type Theorem ◽  
Orthodox Semigroup ◽  
Prima Facie ◽  
Direct Products ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Free Objects ◽  
Completely Regular Semigroups ◽  
Definition Of ◽  
Orthodox Semigroups

For regular semigroups, the appropriate analogue of the concept of a variety seems to be that of an e(xistence)-variety, developed by Hall [6,7,8]. A class V of regular semigroups is an e-variety if it is closed under taking direct products, regular subsemigroups and homomorphic images. For orthodox semigroups, this concept has been introduced under the term “bivariety” by Kaďourek and Szendrei [12]. Hall showed that the collection of all e-varieties of regular semigroups forms a complete lattice under inclusion. Further, he proved a Birkhoff-type theorem: each e-variety is determined by a set of identities. For e-varieties of orthodox semigroups a similar result has been proved by Kaďourek and Szendrei. At variance with the case of varieties, prima facie the free objects in general do not exist for e-varieties. For instance, there is no free regular or free orthodox semigroup. This seems to be true for most of the naturally appearing e-varieties (except for cases of e-varieties which coincide with varieties of unary semigroups such as the classes of all inverse and completely regular semigroups, respectively). This is true if the underlying concept of free objects is denned as usual. Kaďourek and Szendrei adopted the definition of a free object according to e-varieties of orthodox semigroups by taking into account generalized inverses in an appropriate way. They called such semigroups bifree objects. These semigroups satisfy the properties one intuitively expects from the “most general members” of a given class of semigroups. In particular, each semigroup in the given class is a homomorphic image of a bifree object, provided the bifree objects exist on sets of any cardinality. Concerning existence, Kaďourek and Szendrei were able to prove that in any class of orthodox semigroups which is closed under taking direct products and regular subsemigroups, all bifree objects exist and are unique up to isomorphism. Further, similar to the case of varieties, there is an order inverting bijection between the fully invariant congruences on the bifree orthodox semigroup on an infinite set and the e-varieties of orthodox semigroups. Recently, Y. T. Yeh [22] has shown that suitable analogues to free objects exist in an e-variety V of regular semigroups if and only if all members of V are either E-solid or locally inverse.

scholarly journals On the lattice of varieties of completely regular semigroups

1983 ◽  
Vol 35 (2) ◽  
pp. 227-235 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Lattice Of Varieties ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Variety Of Groups ◽  
Completely Regular Semigroups ◽  
Subdirect Products

AbstractSeveral morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).

scholarly journals Complete regularity of Ellis semigroups of -actions

2020 ◽  
pp. 1-17
Author(s):  
MARCY BARGE ◽  
JOHANNES KELLENDONK
Keyword(s):  
Complete Regularity ◽  
Completely Regular ◽  
Ellis Semigroup ◽  
Totally Disconnected

Abstract It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

scholarly journals Two semigroups of continuous relations

1977 ◽  
Vol 23 (1) ◽  
pp. 46-58 ◽  
Author(s):  
A. R. Bednarek ◽  
Eugene M. Norris
Keyword(s):  
Large Class ◽  
Algebraic Structure ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Topological Semigroups ◽  
Stone Type

SynopsisIn this paper we define two semigroups of continuous relations on topological spaces and determine a large class of spaces for which Banach-Stone type theorems hold, i.e. spaces for which isomorphism of the semigroups implies homeomorphism of the spaces. This class includes all 0-dimensional Hausdorff spaces and all those completely regular Hausdorff spaces which contain an arc; indeed all of K. D. Magill's S*-spaces are included. Some of the algebraic structure of the semigroup of all continuous relations is elucidated and a method for producing examples of topological semigroups of relations is discussed.

Globals of completely regular periodic semigroups

1984 ◽  
Vol 29 (1) ◽  
pp. 365-374 ◽  
Author(s):  
Matthew Gould ◽  
Joseph A. Iskra ◽  
Constantine Tsinakis
Keyword(s):  
Completely Regular

Congruences on ideal nil-extension of completely regular semigroups

1998 ◽  
Vol 43 (5) ◽  
pp. 379-381
Author(s):  
Xueming Ren ◽  
Yuqi Guo ◽  
Jiaping Cen
Keyword(s):  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Regular Semigroups ◽  
Nil Extension

A Note Concerning Completely Regular G δ -Spaces

1957 ◽  
Vol 8 (6) ◽  
pp. 1060
Author(s):  
L. J. Heider
Keyword(s):  
Completely Regular

The coincidence approach to stochastic point processes

1975 ◽  
Vol 7 (1) ◽  
pp. 83-122 ◽  
Author(s):  
Odile Macchi
Keyword(s):  
Poisson Process ◽  
Probability Space ◽  
Point Processes ◽  
Counting Process ◽  
Regular Point ◽  
General Point ◽  
Completely Regular ◽  
Photon Process ◽  
Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

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