Some Studies Of Locally Regular Cryptogroups

Reviewing the history of the algebraic theory of semigroups,completely regular semigroups,as an important kind of regular semigroups,is studied by many people. Like other algebraic theories,the main theme in the algebraic theory of semigroups is the study of classification of semigroups.The theory of the variety of semigroups is the useful tool to discuss the classification of semigroups.In the aspect of the study of the variety within completely regular semigroups,the operator L is a resultful method.In《Completely Regular Semigroups》,M.Petrich and N. R.Reilly gave some interesting conclusions about the operator L.For example,a completely regular semigroup S is locally orthodox,then every one-sided principal ideal is orthodox.And other one is that,a completely regular semigroup S is a normal cryptogroup if and only if S is locally a Clifford semigroup;especially,S is a normal band if and only if S is locally a semilattice.There are not many conclusions about the operator L within completely regular semigroups.In this thesis,we discuss locally regular bands firstly.For the close relationship between bands and cryptogroups,we discuss locally regular cryptogroups.With the work which have been done before,the conclusions about locally regular orthocryptogroups are given at last.There are four chapters in this thesis.In the first chapter,we give some preparations.In the second chapter,we discuss locally regular bands.In the third chapter,we discuss locally regular cryptogroups.Lastly,we discuss locally regular orthocryptogroups.