Studies On The Varieties And Subvarieties Of Some J-trivial Monoids With Involution

In the 1970s,I.Simon established the correspondence between formal languages and semigroups,that is,a language is piecewise testable if and only if it can be recognized by some J-trivial monoid.Since then,J-trivial monoids have become an important research object in formal language theory and semigroup algebra theory.In universal algebra,the finite basis problem of semigroups and the counting problem of subvarieties are the focus and hotspot of semigroup variety theory.In this paper,we mainly study the finite basis problem and subvarieties counting problem for some classes of J-trivial monoids:the monoid of order five A01,Catalan monoids,Kiselman monoids,Boolean matrix monoids and Lee monoids with involutions.The main contents are as follows:In chapter three,we study the finite basis problem for involution semigroups of small order.By establishing a sufficient condition under which an involution semigroup is non-finitely based,we prove that the involution J-trivial monoid of order five(A01,*)is non-finitely based.Further,it is shown that all involution semigroups of order four or less are finitely based.Hence(A01,*)is a non-finitely based involution semigroup with minimum order.Let Cn and Kn(n≥ 2)be the Catalan monoid and Kiselman monoid respectively,which play important roles in combinatorics and representation theory of semigroups.M.V.Volkov et al.have shown that the monoids Cn and Kn are finitely based if and only if n ≤4.In chapter four,we explore the finite basis problems for the Catalan monoid(Cn,*)and Kiselman monoid(Kn,*)with involution.It is shown that(Cn,*)and(Kn,*)are finitely based if and only if n=2]and(Kn,*)satisfies the same identities as(Cn,*)for each n ≤4.But if(Cn,*)and(Kn,*)share the same identities when n≥ 5 is still an open question.Let BRn be the semigroup of all Boolean n×n matrices with Is on the main diagonal under the ordinary matrix multiplication.The semigroup BRn admits natural unary operations:the transposition τ and the skew transpositionσ,which make(BRn,τ)and(BRn,σ)involutory semigroups.It is known that the semigroup BRn is finitely based if and only if n ≤4.In chapter five,we prove that(BRn,τ)is finitely based if and only if n≤4,which answers an open question posed by Auinger et al.Further,it is shown that(BRn,σ)is finitely based if and only if n≤2.Therefore,when n≤4,the finite basis properties of(BRn,τ)and(BRn,σ)are different although they share the same semigroup reduct.The Lee monoid Lk1(k≥ 2)plays an important role in the study of semigroup varieties and involution semigroup varieties.O.B.Sapir et al.have shown that the monoid Lk1 is non-finitely based if and only if k≥ 3.The main goal of chapter six is to study the finite basis problem of(Lk1,*).It is shown that(Lk1,*)is nonfinitely based for all odd k≥ 3.Combining with the existing results,we conclude that(Lk1,*)is non-finitely based for all k≥ 2.In chapter seven,we investigate the subvarieties counting problem for involution semigroup varieties.By giving a sufficient condition under which an involution monoid generates a variety with continuum many subvarieties,several involution J-trivial monoids including Rees quotients of free monoids,Lee monoids,Catalan monoids,Kiselman monoids,and Boolean matrix monoids are shown to generate varieties with continuum many subvarieties.