Algebras and ordered sets: structure, enumerability, decidability

9971352 McKenzie McKenzie将研究涉及由它们生成的簇和拟簇确定的有限代数的性质的算法问题：有没有一个算法来确定有限代数A中有效的拟方程是否有限基？是否有一个算法来确定由A生成的拟簇是否具有自然对偶？具有有限等式基的有限代数类是递归可枚举的吗？生成剩余大变量的有限代数类是递归可数的吗？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簇。Farley在对偶理论和集合的组合学中的各种问题上都做出了富有成效的工作。最重要的是一个猜想，一个链完全序集的所有完美序列都收敛到相同的‘核心’，直到同构，以及与此密切相关的猜想，即不动点性质在(无限制的)序集的直积下是保持的。Farley和McKenzie将在指数可拆的情况下证明Birkhoff关于有限序集的指数的消去猜想。普适代数是处理未知事物的数学的一个分支。古语说，如果它看起来像橙子，闻起来像橙子，吃起来像橙子，那么它就是橙子，这是不适用的。一个人可以拥有一个对象的集合，我们所知道的就是关于这些对象的所有人都可以像数字一样对它们进行加法和乘法，并且“通常”规则适用-例如，x+y总是等于y+x-但仍然不处理数字。给定一个特定的数学系统，称之为A，将会有各种各样的数学系统满足相同的代数规则。麦肯齐将考察，如果一个人只知道a，他可以对这种系统说些什么。二、有序集合论是数学的一个分支，研究的是秩序：如果一个人有一个对象的集合，问“对象P比对象Q大吗？”然后是一个哈桑有序集。例如，自然数集合N(1，2，3，4，...)是一个有序集合。有序集可以像数字一样相加和相乘-N*N是点对(a，b)的集合，其中a和b是自然数。一个自然的问题是：如果P和Q是具有特定性质的有序集，那么P*Q是否具有这种性质？当性质是“定点性质”时，Farley将解决这个问题；这是一个长期存在的难题。法利和麦肯齐还将尝试解决一个与有序集的求幂有关的问题，这个问题已经有50多年的历史了。

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.***