Algebras and ordered sets: structure, enumerability, decidability

小行星9971352 McKenzie将调查算法问题，涉及有限代数的性质所确定的品种和准品种，他们产生：是否有一个算法，以确定是否准方程有效的有限代数A是基于代数？是否有一个算法来确定A生成的拟簇是否具有自然对偶性？ 具有有限方程基的有限代数类是递归可递归的吗？ 生成剩余大簇的有限代数类是递归可积的吗？ 此外，他将致力于一个完全确定的所有numbergenerated品种，拥有可定义的主同余。 法利是工作富有成效的各种问题的对偶理论和组合oforded集。 最重要的是猜想，一个链完备有序集的所有完美序列收敛到同一个“核”，直到同构，以及密切相关的猜想，不动点性质在（无限制的）有序集的直积下保持不变。 法利和麦肯齐将试图证明Birkhoff关于有限序集的指数的消去猜想，其中指数是可消去的。 I. 泛代数是数学的一个分支，它研究的是未知数。 古老的格言，如果它看起来像一个橙子，闻起来像一个橙子，尝起来像一个橙子，那么它就是一个橙子，并不适用。 一个人可以有一个对象的集合，我们所知道的是，一个人可以把它们像数字一样相加和相乘，并且“通常”规则适用-例如，x+y总是等于y+x -但仍然不是处理数字。 给定一个特定的代数系统，称之为A，会有各种各样的代数系统满足相同的代数规则。 McKenzie将考察如果一个人只知道A，他能对这种系统说些什么。 二. 有序集理论是数学的一个分支，它处理的是有序：如果一个人有一个对象的集合，问“对象P比对象Q大吗？''然后一个有一个有序集。 例如，自然数（1，2，3，4，.）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是点对（a，B）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，那么P* Q是否具有该性质？ 法利将解决这个问题时，属性是“不动点属性”，这是一个困难的长期存在的问题. 法利和麦肯齐还将试图解决一个关于有序集的幂运算的问题，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.***