Distributivity and permutability of congruence relations in equational classes of algebras
Distributivity and permutability of congruence relations in equational classes of algebras
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代数方程类中同余关系的分布性和置换性
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发表时间：
1963
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通讯作者：
A. Pixley
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作者：
A. Pixley
1。初步概念。在略有不同的术语和符号中，本节中讨论的所有概念和结果（定理1除外）都可以在伯克霍夫[2; 3]。令t为任意，固定的索引集，让S为t的函数，对正整数的集合。如果2 {是一个数学系统5T =（a，k），则TF是物种S的代数，由非空置a和a acet的原始操作a和a家族组成，其中oagaas（）。我们用“物种”代数的s（a） - 元件运行，观察到OA在不同代数中的解释将有所不同。考虑到这一点，我们有以下定义。 S-表达是与OA不确定的不确定或有限组成。 S-身份是方程4）= 4其中4和4'是S表达。如果每个（单个值）分配A到4或4'以4或4'的不确定性分配（单个值）分配，则物种S的代数2满足S-身份4 = 41，则满足4或4'的元素分配，4'，4'的成员a as 4'的同一成员。让你成为任何S-身份。 c（u），我们表示满足u的物种的方程类别，由s的代数组成，这些代数满足所有）= 4，c U.由身份组成：（i）xuy = yux，xg'y = yg'x，（ii）xuj（y“ jz）=（xuy）uz，xfl（y（z）= （xcy）c \ z，（iii）xu（xny）= xc（x.juy）= x，然后c（u）由所有晶格的类组成。 - 身份，我们表示由＆k（u）物种的自由代数，具有k发电机，由身份U.？k（u）确定为k不确定的元素性别类别的元素类别如果两个表达式4和4'放在同一类中，如果4 = 4'是身份的逻辑结果。它可以显示[2]，在续集中，我们要求！k（u） c（u）。
1. Preliminary concepts. In slightly different terminology and notation, all of the concepts and results discussed in this section (except Theorem 1) are to be found in the work of Birkhoff [2; 3]. Let T be an arbitrary, fixed, index set and let S be a function on T to the set of positive integers. tf is an algebra of species S if 2{ is a mathematical system 5t = (A, K) consisting of a nonempty set A and a family K of primitive operations oa, aCET, where oaGAAS (). We denote by ?a the S(a)-ary operation of any algebra of species S, observing that Oa will have a different interpretation in different algebras. With this in mind we have the following definitions. An S-expression is either an indeterminate or a finite composition of indeterminates with the oa. An S-identity is an equation 4) =4 where 4 and 4' are S-expressions. An algebra 2 of species S satisfies the S-identity 4=41 if for each (single valued) assignment of elements of A to the indeterminates appearing in 4 or 4', 4 yields the same member of A as 4'. Let U be any set of S-identities. By C( U) we denote the equational class of species S satisfying U consisting of those algebras of species S which satisfy all of the )=4, C U. For example, if U and n are taken as the primitive operations, and U consists of the identities: (i) xUy = yUx, xG'y = yG'x, (ii) xuJ(y"Jz) = (xuy) uz, xfl(y(z) = (xCy)C\z, (iii) xU(xny) =xC(x.JUy) = x, then C(U) consists of the class of all lattices. If k is any cardinal number and U is any set of S-identities, we denote by &k(U) the free algebra of species S, having k generators, determined by the identities U. ?k(U) has as elements classes of Sexpressions in k indeterminates where two expressions 4 and 4' are placed in the same class if 4 =4' is a logical consequence of the identities U. It can be shown [2], and in the sequel we require the fact, that !k( U) C( U).