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Some relations on the lattice of varieties of completely regular semigroups

Mario Petrich — 2002

Bollettino dell'Unione Matematica Italiana

On the lattice L ⁢ C ⁢ R of varieties of completely regular semigroups considered as algebras with the binary multiplication and unary inversion within maximal subgroups, we study the relations K l , K , K r , T l , T , T r , C and L . Here K is the kernel relation, T is the trace relation, T l and T r are the left and the right trace relations, respectively, K p = K ∩ T p for p ∈ l , r , C is the core relation and L is the local relation. We give an alternative definition for each of these relations P of the form U P V ⇔ U ∩ P ~ = V ∩ P ~ ( U , V ∈ L ( C R ) ) , for some subclasses P ~ of C ⁢ R ....

Certain partial orders on semigroups

Mario Petrich — 2001

Czechoslovak Mathematical Journal

Relations introduced by Conrad, Drazin, Hartwig, Mitsch and Nambooripad are discussed on general, regular, completely semisimple and completely regular semigroups. Special properties of these relations as well as possible coincidence of some of them are investigated in some detail. The properties considered are mainly those of being a partial order or compatibility with multiplication. Coincidences of some of these relations are studied mainly on regular and completely regular semigroups.

On sandwich sets and congruences on regular semigroups

Mario Petrich — 2006

Czechoslovak Mathematical Journal

Let S be a regular semigroup and E ( S ) be the set of its idempotents. We call the sets S ( e , f ) f and e S ( e , f ) one-sided sandwich sets and characterize them abstractly where e , f ∈ E ( S ) . For a , a ' ∈ S such that a = a a ' a , a ' = a ' a a ' , we call S ( a ) = S ( a ' a , a a ' ) the sandwich set of a . We characterize regular semigroups S in which all S ( e , f ) (or all S ( a ) ) are right zero semigroups (respectively are trivial) in several ways including weak versions of compatibility of the natural order. For every a ∈ S , we also define E ( a ) as the set of all idempotets e such that, for any congruence ρ on...

Characterizing pure, cryptic and Clifford inverse semigroups

Mario Petrich — 2014

Czechoslovak Mathematical Journal

An inverse semigroup S is pure if e = e 2 , a ∈ S , e < a implies a 2 = a ; it is cryptic if Green’s relation ℋ on S is a congruence; it is a Clifford semigroup if it is a semillatice of groups. We characterize the pure ones by the absence of certain subsemigroups and a homomorphism from a concrete semigroup, and determine minimal nonpure varieties. Next we characterize the cryptic ones in terms of their group elements and also by a homomorphism of a semigroup constructed in the paper. We also characterize groups and...

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