Algebras and ordered sets: structure, enumerability, decidability

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9971352McKenzie McKenzie will investigate algorithmic questions that involveproperties of finite algebras determined by the varieties andquasi-varieties they generate: Is there an algorithm to determine ifthe quasi-equations valid in a finite algebra A are finitely based?Is there an algorithm to determine if the quasi-variety generatedby A possesses a natural duality? Is the class of finite algebraspossessing a finite equational basis recursively enumerable? Is theclass of finite algebras generating a residually large varietyrecursively enumerable? Additionally, he will work toward a completedetermination of all finitely generated varieties that possessdefinable principal congruences. Farley is working productively on alarge variety of problems in duality theory and combinatorics ofordered sets. Most significant is the conjecture that all perfectsequences of a chain-complete ordered set converge to the same``core,'' up to isomorphism, and, closely related, the conjecture thatthe fixed-point property is preserved under direct products of(unrestricted) ordered sets. Farley and McKenzie will attempt toprove Birkhoff's cancellation conjecture for exponents of finiteordered sets in the case where the exponent is dismantlable. I. Universal algebra is a branch of mathematics that deals withthe unknown. The old adage, that if it looks like an orange, smellslike an orange, and tastes like an orange, then it IS an orange, doesnot apply. One can have a collection of objects about which all weknow is that one can add and multiply them like numbers, and that the``usual'' rules apply --- for instance, x+y always equals y+x --- butstill not be dealing with numbers. Given a particular mathematicalsystem, call it A, there will be a whole variety of mathematicalsystems that satisfy the same algebraic rules. McKenzie will examinewhat one can say about this variety of systems if one just knows aboutA. II. The theory of ordered sets is a branch of mathematics thatdeals with, well, order: If one has a collection of objects, and itmakes sense to ask, ``Is object P bigger than object Q?'' then one hasan ordered set. For instance, the collection N of natural numbers(1,2,3,4,...) is an ordered set. Ordered sets can be added andmultiplied like numbers --- N*N is the collection of pairs of points(a,b) where a and b are natural numbers. A natural question to askis: If P and Q are ordered sets with a particular property, does P*Qhave this property? Farley will address this question when theproperty is the "fixed point property"; this is a difficultlong-standing problem. Farley and McKenzie will also attempt to solvea problem dealing with the exponentiation of ordered sets, a problemover 50 years old.***