On The Finite Basis Problem For Some Monoids

In this thesis, we study the ?nite basis problem for some monoids and obtain some new meaningful results.Let Knbe the Kauffman monoid generanted by n(n ≥ 2) generators. In Chapter three, we prove a su?cient condition for the non-?nite basis property of semigroups.As applications, it is shown that both K3 and K3\ {1} are non-?nitely based. At the end of the chapter, some further applications of the su?cient condition will be given.Let Mn(T) be the semigroup of all n × n matrices with entries in the tropical semiring T. The monoid of all upper(resp. lower) triangular tropical matrices is denoted by Un(T)(resp. Ln(T)). In [29], Izhakian and Margolis proved that there are non-trivial identities satis?ed by U2(T) and M2(T). Later, by the correspondence between tropical matrices and weighted digraphs, Izhakian proved that the monoid of all n × n triangular tropical matrices satis?es a nontrivial identity [30]. In Chapter four, the ?nite basis problem for U2(T) will be investigated. By giving a new su?cient condition under which a semigroup is non-?nitely based, it is shown that the monoid of 2 × 2 upper triangular tropical matrices is non-?nitely based.Let Cnbe the Chinese monoid of rank n. In [34], Jaszu′nska and Okni′nski proved that Cnembeds into the product Bi× Zj, for some positive integers i, j depending on n, where B denotes the bicyclic monoid. So, Cnsatis?es the identities hold in B. In Chapter ?ve, it will be investigated that the ?nite basis problem for Cn. By a su?cient condition under which a semigroup is non-?nitely based, we prove that when n > 1, Cnis non-?nitely based. Since it is clear that C1 is ?nitely based, we give a complete answer to the ?nitely basis problem for the Chinese monoid.Let Tn(F) and U Tn(F) be the semigroups of all upper triangular n × n matrices and all upper triangular n × n matrices with 0s and/or 1s on the main diagonal over a ?eld F, respectively. In [92], Volkov proved that, U T3(R), both as the plain semigroup and as the involution semigroup under the skew transposition, is non-?nitely based. In Chapter six, it is shown that for any ?eld F, U T2(F) is ?nitely based and a ?nite identity basis for U T2(F) will be given. By giving a su?cient condition under which an involution semigroup is non-?nitely based, we will show that both T2(F) and U T2(F) as the involution semigroups under the skew transposition are non-?nitely based, and there is a continuum non-?nitely based involution monoid varieties between the involution monoid variety Var(U T2(F)) generated by U T2(F)and the involution monoid variety Var(T2(F)) generated by T2(F), in which char(F) =0. Moreover, Var(U T2(F)) cannot be de?ned within Var(T2(F)) by any ?nite set of identities.Let U Tn(F) and U T±n(F) be the semigroup constituted by the triangular n ×n matrices over the ?eld F whose entries taking values in {-1, 1} and {-1, 0, 1},respectively. In Chapter seven, we give two new su?cient conditions under which a semigroup is non-?nitely based. And it is shown that if char(F) = 0, then both U T-2(F) and U T±2(F) are non-?nitely based.