scholarly journals Quadratic level quasigroup equations with four variables II: The Lattice of varieties

2013 ◽  
Vol 93 (107) ◽  
pp. 29-47 ◽  
Author(s):  
Aleksandar Krapez
Keyword(s):  
Lattice Of Varieties ◽  
Operation Symbol

We consider a class of quasigroup identities (with one operation symbol) of the form x1x2?x3x4=x5x6?x7x8 and with xi?{x, y, u, v} (1? i ? 8) with each of x, y, u, v occurring exactly twice in the identity. There are 105 such identities. They generate 26 quasigroup varieties. The lattice of these varieties is given.

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 Certain characterizations of varieties of bands

1988 ◽  
Vol 31 (2) ◽  
pp. 301-319 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Rectangular Band ◽  
Simple System ◽  
Left Zero ◽  
Green’S Relations ◽  
Transformation Semigroups ◽  
Lattice Of Varieties ◽  
Green's Relations ◽  
The World ◽  
New System

The lattice of varieties of bands was constructed in [1] by providing a simple system of invariants yielding a solution of the world problem for varieties of bands including a new system of inequivalent identities for these varieties. References [3] and [5] contain characterizations of varieties of bands determined by identities with up to three variables in terms of Green's relations and the functions figuring in a construction of a general band. In this construction, the band is expressed as a semilattice of rectangular bands and the multiplication is written in terms of functions among these rectangular band components and transformation semigroups on the corresponding left zero and right zero direct factors.

Completely simple semigroups: free products, free semigroups and varieties

1981 ◽  
Vol 88 (3-4) ◽  
pp. 293-313 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Matrix Representation ◽  
Maximal Subgroups ◽  
Lattice Of Varieties ◽  
Free Products ◽  
Completely Simple Semigroups ◽  
Countable Rank ◽  
Free Semigroups ◽  
Abelian Subgroups ◽  
And Inversion ◽  
Completely Simple

SynopsisThe class CS of completely simple semigroups forms a variety under the operations of multiplication and inversion (x−1 being the inverse of x in its ℋ-class). We determine a Rees matrix representation of the CS-free product of an arbitrary family of completely simple semigroups and deduce a description of the free completely simple semigroups, whose existence was proved by McAlister in 1968 and whose structure was first given by Clifford in 1979. From this a description of the lattice of varieties of completely simple semigroups is given in terms of certain subgroups of a free group of countable rank. Whilst not providing a “list” of identities on completely simple semigroups it does enable us to deduce, for instance, the description of all varieties of completely simple semigroups with abelian subgroups given by Rasin in 1979. It also enables us to describe the maximal subgroups of the “free” idempotent-generated completely simple semigroups T(α, β) denned by Eberhart et al. in 1973 and to show in general the maximal subgroups of the “V-free” semigroups of this type (which we define) need not be free in any variety of groups.

THE VARIETY OF RINGS WITH INVOLUTION SATISFYING x7≈ x

2008 ◽  
Vol 01 (03) ◽  
pp. 397-414
Author(s):  
Tiwadee Musunthia
Keyword(s):  
Galois Field ◽  
Lattice Of Varieties ◽  
Rings With Involution ◽  
Variety Of Rings

We give a complete description of the lattice of varieties of rings with involution satisfying x7≈ x by identity bases. There are 90 such varieties. If we substitute in each ring of such a variety the operations by term operations of the same arity we obtain a so-called class of derived rings. We discuss the case when the class of derived rings belongs to the original variety. In particular, we describe the class of derived rings for the variety of rings generated by the two-element Galois-field.

scholarly journals The Lattice of Varieties of Implication Semigroups

Order ◽  
2019 ◽  
Vol 37 (2) ◽  
pp. 271-277 ◽  
Author(s):  
Sergey V. Gusev ◽  
Hanamantagouda P. Sankappanavar ◽  
Boris M. Vernikov
Keyword(s):  
Lattice Of Varieties

Infinite Distributivity in the Lattice of Varieties of m-groups

2015 ◽  
Vol 54 (1) ◽  
pp. 1-9
Author(s):  
N. V. Bayanova ◽  
A. V. Zenkov
Keyword(s):  
Lattice Of Varieties ◽  
Infinite Distributivity

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.

On the Lattice of Varieties of Involution Semigroups

2001 ◽  
Vol 62 (3) ◽  
pp. 438-459 ◽  
Author(s):  
Igor Dolinka
Keyword(s):  
Lattice Of Varieties
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