scholarly journals A note on congruences on orthodox semigroups

1985 ◽  
Vol 26 (1) ◽  
pp. 25-30
Author(s):  
D. B. McAlister
Keyword(s):  
Inverse Semigroup ◽  
Orthodox Semigroup ◽  
Explicit Construction ◽  
Lattice Of Congruences ◽  
Orthodox Semigroups

C. Eberhart and W. Williams [3] showed that the least inverse semigroup congruence , on an orthodox semigroup S, plays a very important role in determining the structure of the lattice of congruences on S. In this note we show that their results can be applied to give an explicit construction for the idempotent separating congruences on S in terms of idempotent separating congruences on S/.

On left quasinormal orthodox semigroups

1983 ◽  
Vol 95 (1-2) ◽  
pp. 59-71 ◽  
Author(s):  
Gracinda M. S. Gomes
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Orthodox Semigroup ◽  
Local Isomorphism ◽  
Orthodox Semigroups

SynopsisThe existence of a smallest inverse congruence on an orthodox semigroup is known. It is also known that a regular semigroup S is locally inverse and orthodox if and only if there exists a local isomorphism from S onto an inverse semigroup T.In this paper, we show the existence of a smallest R-unipotent congruence ρ on an orthodox semigroup S and give its expression in the case where S is also left quasinormal. Finally, we prove that a regular semigroup S is left quasinormal and orthodox if and only if there exists a local isomorphism from S onto an R-unipotent semigroup T.

scholarly journals Semigroups whose idempotents form a subsemigroup

1990 ◽  
Vol 41 (2) ◽  
pp. 161-184 ◽  
Author(s):  
Jean-Camille Birget ◽  
Stuart Margolis ◽  
John Rhodes
Keyword(s):  
Inverse Semigroup ◽  
Finite Semigroup ◽  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Type Ii ◽  
Finite Semigroups ◽  
Case Type ◽  
Orthodox Semigroups

We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.For these three classes of semigroups, type-II is equal to type-II construct.

scholarly journals Congruences on an orthodox semigroup via the minimum inverse semigroup congruence

1977 ◽  
Vol 18 (2) ◽  
pp. 181-192 ◽  
Author(s):  
Carl Eberhart ◽  
Wiley Williams
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Orthodox Semigroup ◽  
Band Congruence ◽  
Lattice Of Congruences

It is well known that the lattice Λ(S) of congruences on a regular semigroup S contains certain fundamental congruences. For example there is always a minimum band congruence β, which Spitznagel has used in his study of the lattice of congruences on a band of groups [16]. Of key importance to his investigation is the fact that β separates congruences on a band of groups in the sense that two congruences are the same if they have the same meet and join with β. This result enabled him to characterize θ-modular bands of groups as precisely those bands of groups for which ρ⃗(ρ∨β, ρ∧β)is an embedding of Λ(S) into a product of sublattices.

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 On regular semigroups whose idempotents form a subsemigroup

1969 ◽  
Vol 1 (2) ◽  
pp. 195-208 ◽  
Author(s):  
T. E. Hall
Keyword(s):  
Inverse Semigroup ◽  
Simple Form ◽  
Finite Semigroup ◽  
Orthodox Semigroup ◽  
Principal Factor ◽  
Regular Semigroups ◽  
Orthodox Semigroups

For brevity the semigroups in the title are called orthodox semigroups. The finest inverse semigroup congruence on an orthodox semigroup is shown to have a simple form and conversely, regular semigroups whose finest inverse congruence has this simple form are shown to be orthodox. Next ideal extensions of orthodox semigroups by orthodox semigroups are shown to be also orthodox, whence a finite semigroup is orthodox if and only if each principal factor is orthodox and completely O-simple or simple. Finally it is determined which completely O-simple semigroups are orthodox.

scholarly journals On quasi-F-orthodox semigroups

1995 ◽  
Vol 38 (3) ◽  
pp. 361-385 ◽  
Author(s):  
Bernd Billhardt ◽  
Mária B. Szendrei
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Semidirect Product ◽  
Orthodox Semigroup ◽  
Band Variety ◽  
Orthodox Semigroups

An orthodox semigroup S is termed quasi-F-orthodox if the greatest inverse semigroup homomorphic image of S1 is F-inverse. In this paper we show that each quasi-F-orthodox semigroup is embeddable into a semidirect product of a band by a group. Furthermore, we present a subclass in the class of quasi-F-orthodox semigroups whose members S are embeddable into a semidirect product of a band B by a group in such a way that B belongs to the band variety generated by the band of idempotents in S. In particular, this subclass contains the F-orthodox semigroups and the idempotent pure homomorphic images of the bifree orthodox semigroups.

XII.—The Lattice of Congruences on a Bisimple ω-Semigroup

1967 ◽  
Vol 67 (3) ◽  
pp. 175-184
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Natural Numbers ◽  
Necessary And Sufficient ◽  
Lattice Of Congruences

SynopsisA necessary and sufficient condition is determined for the modularity of the lattice of congruences on a bisimple inverse semigroup whose semilattice of idempotents is order-anti-isomorphic to the set of natural numbers.

scholarly journals The Least Commutative Congruence on a simple regular ω-semigroup

1990 ◽  
Vol 32 (1) ◽  
pp. 13-23 ◽  
Author(s):  
C. Bonzini ◽  
A. Cherubini ◽  
B. Piochi
Keyword(s):  
Inverse Semigroup ◽  
Explicit Construction ◽  
Inverse Semigroups

Piochi in [10] gives a description of the least commutative congruence γ of an inverse semigroup in terms of congruence pairs and generalizes to inverse semigroups the notion of solvability. The object of this paper is to give an explicit construction of λ for simple regular ω-semigroups exploiting the work of Baird on congruences on such semigroups. Moreover the connection between the solvability classes of simple regular ω-semigroups and those of their subgroups is studied.

JOINS OF INVERSE SEMIGROUP VARIETIES AND BAND VARIETIES

1991 ◽  
Vol 01 (03) ◽  
pp. 371-385 ◽  
Author(s):  
PETER R. JONES ◽  
PETER G. TROTTER
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Lattice Of Varieties ◽  
Orthodox Semigroups ◽  
Semigroup Varieties

The joins in the title are considered within two contexts: (I) the lattice of varieties of regular unary semigroups, and (II) the lattice of e-varieties (or bivarieties) of orthodox semigroups. It is shown that in each case the set of all such joins forms a proper sublattice of the respective join of the variety I of all inverse semigroups and the variety B of all bands; each member V of this sublattice is determined by V ∩ I and V ∩ B. All subvarieties of the join of I with the variety RB of regular bands are so determined. However, there exist uncountably many subvarieties (or sub-bivarieties) of the join I ∨ B, all of which contain I and all of whose bands are regular.

scholarly journals On right unipotent semigroups II

1978 ◽  
Vol 19 (1) ◽  
pp. 63-68 ◽  
Author(s):  
P. S. Venkatesan
Keyword(s):  
Inverse Semigroup ◽  
Orthodox Semigroup

We describe two congruences α and γ contained in ℒ on an arbitrary orthodox semigroup. Let S be a right unipotent semigroup. We show that (i) α is an inverse semigroup congruence and γ is the finest fundamental inverse semigroup congruence on S, (ii) S is a union of groups if and only if ℒ on S and (iii) S is a band of groups if and only if ℒ on S.

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