Some Studies On The Finite Basis Problem For Semigroups

An algebra is said to be finitely based if every identity it satisfies can be derived from a finite subset of the set of identities it satisfies. Otherwise it is called non-finitely based. The finite basis problem for finite algebras is a classical research subject in the theory of universal algebra. It has been shown that some classical algebras, such as finite groups, finite rings, finite lattices and finite Lie-algebras, are finitely based. However, the finite basis problem for finite semigroups is still open. In this thesis, using the method of syntactic analysis, we do some research on the finite basis problem for a class of the transformation semigroups over a chain of order n, the semigroup of 2 x 2 upper triangular matrices (over fields, semirings, distributive lattices), and some generalized Rees matrix semigroups, and obtain some meaningful results. The details are as follows:Denote by PEIn (POEIn) the monoid of partial (order-preserving) extensive injective transformations over a chain of order n. In Chapter three, a sufficient condi-tion for non-finitely based semigroups is introduced. As an application of the sufficient condition, it is shown that PEI%(POEI3) is non-finitely based. Furthermore, this together with the results of Edmunds and Goldberg gives a complete answer to the finite basis problem and the hereditary finite basis problem for the monoid PEIn (POEIn), that is, it is proven that the monoid PEIn (POEIn) is non-finitely based if and only if n≥ 3; the monoid PEIn (POEIn) is hereditarily finitely based if and only if n< 2.Denote by UTn(F) the semigroup of all n x n upper triangular matrices over a field F whose main diagonal entries are Os or 1s with n≥ 2. For the real number field R, Volkov proved that the semigroup UT3(R) is non-finitely based. When F is the two-element field, Zhang, Li and Luo verified that UT2(F) is hereditarily finitely based. Chen et al. showed that for any field F, the semigroup UT2(F) is finitely based, and a finite identity basis for UT2(F) is given. For any field F whose characteristic is not equal to 0, the finite basis problem for the subvarieties of the variety var(UT2(F)) is investigated in Chapter four. It is shown that the semigroup UT2(F) is hereditarily finitely based.Let L be a distributive lattice with 0 and 1, TMn(L) be the semigroup of all n × n upper triangular matrices over L. Denote by UMn(T) the semigroup of all n×n upper triangular matrices over the tropical semiring T = RU{ -∞} whose main diagonal entries are 0s or -∞s. In Chapter five, a sufficient condition for finitely based semigroups is presented. By the sufficient condition, it is shown that the semigroups TM2(L) and UM2(T) are finitely based, respectively, so the finite basis problem for them is solved.Let Cm,r1 be a monogenic monoid, RM(ai) be the generalized Rees matrix semi-group M0(Cm,r1,{1,2},{1,2}, (0 1 1 ai)), RMi be the generalized Rees matrix semigroup M0(Cm,r1,{1,2},{1,2}, (0 anr+r anr+r ai)), in which anr+r is the unique idempotent of the monogenic monoid Cm,r. In Chapter six, a sufficient condition for non-finitely based semigroups is given. Using the sufficient condition and some known results, we dis-cuss and solve the finite problem for RM(ai) and RMi, respectively, where m, r≥ 1, 0≤ i≤ m+r - 1. As another application of the sufficient condition, it is shown that a transformation semigroup of order 8 over a chain of order 4 is non-finitely based.