Some Studies On Varieties Generated By Finite Semigroups And Their Subvarieties Lattices

This thesis does some researches on varieties generated by finite semigroups of small orders and their subvarieties lattice, obtains some new meaningful results, and consists of seven chapters.In Chapter one, the background and major results on semigroup varieties, and the main results of this thesis are introduced. In Chapter two, some fundamental notations, definitions and terminologies that will be used in this thesis are presented.Denote by S_n the variety generated by all semigroups of order n. In [17], M. Jackson proved that the variety S_n(n≥4) contains uncountably many subvarieties. It is easy to show that the variety S₂ contains precisely 32 subvarieties.M. Jackson [17] posed the question: whether or not the variety S₃ contains uncountably many subvarieties? In Chapter three, the subvarieties of the variety S₃ will be investigated. It is shown that the variety S₃ is hereditarily finitely based, and so the variety S₃ contains countably infinitely many subvarieties, which answers M. Jackson's question in negative.A semigroup is complex if it generates a variety the subvariety lattice of which contains an isomorphic copy of every finite lattice. In [30], E. W. H. Lee has verified that there are precisely 4 distinct minimal complex semigroups, denoted by S₄²², S₄²¹,S₄²³, and S₅². Descriptions of subvarieties lattices of varieties generated by S₄²² and S₄²¹ respectively follow from the results of [28] and [29]. In Chapter four, all subvarieties of varieties S₄²³ and S₅² generated by S₄²³ and S₅² respectively will be described. Note that S₄²³ is dual to a subvariety of variety S₅². In essence, we described all subvarieties of S₅², and did some research on its subvarieties lattice.A semigroup is said to be finitely based if it generates a variety which is finitely based. Since a finite semigroup is finitely based if it is either permutative or idempotent, C. C. Edmunds [8] proved that all semigroups of order four that are neither permutative nor idempotent are finitely based. There are ten such semigroups up to isomorphism and anti-isomorphism. Descriptions of all subvarieties of varieties generated by eight of these ten semigroups follow from [14], [28], [29] and [63]. In Chapter five, the join S₄¹¹ V S₄²⁵ of the variety generated by the remaining two semigroups S₄¹¹ and S₄²⁵ will be investigated. All subvarieties of S₄¹¹ V S₄²⁵ are finitely based and finitely generated and their identity bases and generating semigroups will be given. The lattice (?)(S₄¹¹ V S₄²⁵) of subvarieties of S₄¹¹ V S₄²⁵ is also completely described. Specifically, the lattice (?)(S₄¹¹ V S₄²⁵) is finite and non-modular.In [60], M. V. Volkov has shown that for each integer n > 4, the variety generated by all finitely based semigroups of order n is nonfinitely based. It is well known that each variety generated by semigroups of order n (n≤5) is finitely based. Therefore for each integer n≤4, the variety generated by all finitely based semigroups of order n is equal to the variety S_n generated by all semigroups of order n. It is known that the variety S₂ and S₃ are finitely based. In Chapter six, it will be investigated that the finite basis problem for the variety S'₄ generated by all non-permutative and non-idempotent semigroups of order four. It is shown that the variety S'₄ is finitely based and a finite identity basis will be given.In [15], I. A. Goldberg defined a transformation monoid E'_n, proved that the monoid E'_n is nonfinitely based if n≥3. The finite basis problem of the variety E'₂ is still open. In Chapter seven, it will be shown that E'₂ is finitely based and a finite identity basis will be given, which solves the finite basis problem for the monoid E'₂. Moreover, it is shown that xyx is not an isoterm for E'₂ and the variety E'₂ generated by E'₂ has uncountably many subvarieties, which gives an affirmative answer to one of M. Jackson's question.