caa a22 4500 445345209 CHVBK 20180317142826.0 cr unu---uuuuu 170323e20111201xx s 000 0 eng 10.1007/s00233-010-9287-1 doi (NATIONALLICENCE)springer-10.1007/s00233-010-9287-1 Petrich Mario 21 420 Bol, Brač, Croatia aut On some medial semigroups with an associate subgroup [Elektronische Daten] [Mario Petrich] Let S be a semigroup and s,t∈S. We say that t is an associate of s if s=sts. If S has a maximal subgroup G such that every element s of S has a unique associate in G, say s ∗, we say that G is an associate subgroup of S and consider the mapping s→s ∗ as a unary operation onS. In this way, semigroups with an associate subgroup may be identified with unary semigroups satisfying three simple axioms. Among them, only those satisfying the identity (st)∗=t ∗ s ∗, called medial, have a structure theorem, due to Blyth and Martins. We introduce several relations germane to the presence of this unary operation. Next we characterize medial, orthodox, cryptic and medial, and cryptic and orthodox semigroups with an associate subgroup in terms of a construction and several choices of a basis of their identities. In addition, we characterize medial monoids and discuss F-inverse semigroups in terms of a construction and bases for their systems of identities. We establish an isomorphism between the lattice of varieties of cryptic medial monoids and the direct product of the lattice of band monoids and the lattice of group varieties. An embedding into idempotent generated semigroups is proved as well. Springer Science+Business Media, LLC, 2011 Semigroup nationallicence Associate subgroup nationallicence Unary semigroup nationallicence Axiom nationallicence Regular nationallicence Orthodox nationallicence Inverse nationallicence Cryptic nationallicence Overabelian nationallicence Semidirect product nationallicence Direct product nationallicence Variety nationallicence Embedding nationallicence Semigroup Forum Springer-Verlag 83/3(2011-12-01), 412-446 0037-1912 83:3<412 2011 83 233 https://doi.org/10.1007/s00233-010-9287-1 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00233-010-9287-1 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Petrich Mario 21 420 Bol, Brač, Croatia aut NATIONALLICENCE 773 0- Semigroup Forum Springer-Verlag 83/3(2011-12-01), 412-446 0037-1912 83:3<412 2011 83 233 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer