Small semigroup related structures with infinite properties

In mathematics, one frequently encounters constructions of a pathological or critical nature. In this thesis we investigate such structures in semigroup theory with a particular aim of finding small, finite, examples with certain associated infinite characteristics. We begin our investigation with a study of the identities of finite semigroups. A semigroup (or the variety it generates) whose identities admit a finite basis is said to be finitely based. We find examples of pairs of finite (aperiodic) finitely based semigroups whose direct product is not finitely based (answering a question of M. Sapir) and of pairs of finite (aperiodic) semigroups that are not finitely based whose direct product is finitely based. These and other semigroups from a large class (the class of finite Rees quotients of free monoids) are also shown to generate varieties with a chain of finitely generated supervarieties which alternate between being finitely based and not finitely based. Furthermore it is shown that in a natural sense, "almost all" semigroups from this class are not finitely based. Not finitely based semigroups that are locally finite and have the property that every locally finite variety containing them is also not finitely based are said to be inherently not finitely based. We construct all minimal inherently not finitely based divisors in the class of finite semigroups and establish several results concerning a fundamental example with this property; the six element Brandt semigroup with adjoined identity element, B. 1/2. We then find the first examples of finite semigroups admitting a finite basis of identities but generating a variety with uncountably many subvarieties (indeed with a chain of subvarieties with the same ordering as the real numbers). For some well known classes, a complete description of the members with this property are obtained and related examples and results concerning joins of varieties are also found. A connection between these results and the construction of varieties with decidable word problem but undecidable uniform word problem is investigated. Finally we investigate several embedding problems not directly concerned with semigroup varieties and show that they are undecidable. The first and second of these problems concern the fundamental relations of Green; in addition some small examples are found which exhibit unusual related properties and a problem of M. Sapir is solved. The third of the embedding problems concerns the potential embeddability of finite semigroup amalgams. The results are easily extended to the class of rings