The Lattice Of Weakly Inverse Subsemigroups Of A Weakly Inverse Semigroup And The Lattice Of Full Subsemigroups Of A Regular Semigroup

In this paper, the concept of weakly inverse subsemigroup of aweakly inverse semigroup S is defined. It is proved that the lattice subw S of allweakly inverse subsemigroups of S is a complete sublattice of the lattice subSof all subsemigroups of S. Moreover a lattice congruenceγJ on subfw S of fullweakly inverse subsemigroups of S is also defined:AγJB (?) A∩J = B∩J.Then a sufficient and necessary condition for the lattice subfw S to be dis-tributive (modular, semimodular) is given: subfw S is distributive (modular,semimodular), if and only if (?)J∈S/J ,(subfw S)/γJ is distributive (modu-lar,semimodular), if and only if (?)J∈S/J ,[fw id(J) ,fw id(J) ] is distribu-tive (modular,semimodular). It is also proved that [fw id(J) ,fw id(J) ] isa filter of the lattice subfw (PF(J)) of full weakly inverse subsemigroups ofthe principal factor PF(J) of S. Particularly, when S is an inverse semigroup,then [fw id(J) ,fw id(J) ] and subfw (PF(J)) are isomorphic, so that thefactorization theorem for the lattice of full inverse subsemigroup of an inversesemigroup is a corollary of our result.Furthermore, the lattice of all full subsemigroups of a regular semigroupis investigated in a same way. A su?cient and necessary condition for thelattice subf S of all full subsemigroups of a regular semigroup S to be distribu-tive (modular, semimodular) is also obtained: subf S is distributive (modular,semimodular), if and only if (?)J∈S/J ,(subf S)/γJ is distributive (modu- lar, semimodular), if and only if (?)J∈S/J ,[f id(J) ,f id(J) ] is distributive(modular,semimodular). Similarly, [f id(J) ,f id(J) ] is also a filter of thelattice subf (PF(J)) of full subsemigroups of the principal factor PF(J) of S.Finally, as a special case, the factorization theorem for the lattice of fullsubsemigroups of an orthodox semigroup is obtained.