Algebras and ordered sets: structure, enumerability, decidability

9971352McKenzie将研究涉及有限代数的性质的算法问题，这些性质是由它们产生的变量和准变量决定的：是否有一种算法来确定在有限代数a中有效的准方程是有限基的？是否存在一种算法来确定由A生成的拟变量是否具有自然对偶性？有限代数类是否具有有限的等式基递归可枚举？一类有限代数产生一个剩余的大变量递归可枚举吗？此外，他将致力于完全确定所有具有可定义主同余的有限生成的变种。法利在对偶理论和有序集组合学的大量问题上取得了丰硕的成果。最重要的是关于链完备有序集的所有完备序列收敛于同一个“核”直至同构的猜想，以及与之密切相关的关于在（无限制）有序集的直积下不动点性质保持不变的猜想。Farley和McKenzie将尝试在指数可拆解的情况下证明有限有序集合指数的Birkhoff消去猜想。通用代数是研究未知事物的数学分支。老话说，如果它看起来像橘子，闻起来像橘子，尝起来像橘子，那么它就是橘子，这并不适用。我们可以有一组对象，我们所知道的是，我们可以将它们像数字一样相加和相乘，并且适用“通常”规则——例如，x+y总是等于y+x——但仍然不是在处理数字。给定一个特定的数学系统，称之为a，就会有各种各样的数学系统满足相同的代数规则。麦肯齐将研究，如果一个人只知道a，他会对这种不同的系统说些什么。2。有序集理论是数学中处理顺序的一个分支：如果有一个对象的集合，问“对象P大于对象Q吗？”是有意义的。然后一个是有序集合。例如，自然数（1,2,3,4，…）的集合N是一个有序集合。有序集合可以像数字一样相加和相乘——N*N是点（a,b）对的集合，其中a和b是自然数。一个很自然的问题是：如果P和Q是具有特定性质的有序集合，那么P* q有这个性质吗？当属性是“定点属性”时，Farley将解决这个问题；这是一个长期存在的难题。Farley和McKenzie还将尝试解决一个50年前就存在的问题——有序集的幂

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.***