Discriminator varieties of Boolean algebras with residuated operators
Discriminator varieties of Boolean algebras with residuated operators
复制标题
具有剩余算子的布尔代数的判别簇
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发表时间：
1993
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通讯作者：
P. Jipsen
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作者：
P. Jipsen
鉴别代数和簇的理论已经得到了广泛的研究，并为我们提供了适用于这类代数和簇的具体例子的丰富的信息和技术。在这里，我们给出了几个这样的例子，对于带有剩余二元算子的布尔代数，简称为r-代数。更具体地说，我们证明了所有有限r-代数、所有整数r-代数、所有在单位下有有限多个元素的单位r-代数以及所有交换剩余么半群都是鉴别代数，只要它们是次直不可约的。然后利用这些结果给出了某些r-代数簇的等价基。我们还证明了剩余布尔么半群的簇不是鉴别子簇，这回答了B.Jonsson的一个问题。1.预赛。布尔代数A0=(A，+，0，·，1，−)上的一元运算f如果f(x+y)=f(X)+f(Y)是可加的，如果f(0)=0是正规的。对于A0上的n元运算f，序列a∈A和i<；n，我们定义f的(a，i)-平移为一元运算fa，i(X)=f(a0，.。。，ai−1，x，ai+1，.。。，一个−1)。A0上的算子是一个n元运算，它的所有(a，i)-平移都是可加的且正规的。请注意，0元运算(常量)没有转换，因此默认情况下它们是运算符。A=(A0，F)是一个带算子(简称BaO)的布尔代数，如果每个f∈F都是A0上的一个算子。F的大小(或秩次)用%f表示。当然，成为布尔代数上的算子是一种等式性质，所有在F中具有算子的BAO的种类将由BAOF表示。簇Bao{f}，其中f是一元算子，通常被称为模代数簇(模逻辑的代数对应)。1991年数学科目分类：小学06E25、中学03G15、08A40、03B45。这篇论文是最终版本，不会在其他地方发表。
The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral ralgebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated monoids are discriminator algebras, provided they are subdirectly irreducible. These results are then used to give equational bases for some varieties of r-algebras. We also show that the variety of all residuated Boolean monoids is not a discriminator variety, which answers a question of B. Jonsson. 1. Preliminaries. A unary operation f on a Boolean algebra A0 = (A,+, 0, ·, 1,− ) is additive if f(x + y) = f(x) + f(y) and normal if f(0) = 0. For an n-ary operation f on A0, a sequence a ∈ A and i < n we define the (a, i)-translate of f to be the unary operation fa,i(x) = f(a0, . . . , ai−1, x, ai+1, . . . , an−1) . An operator on A0 is an n-ary operation for which all (a, i)-translates are additive and normal. Note that 0-ary operations (constants) have no translates, so they are operators by default. A = (A0,F) is a Boolean algebra with operators (BAO for short) if each f ∈ F is an operator on A0. The arity (or rank) of f is denoted by %f . To be an operator on a Boolean algebra is of course an equational property, and the variety of all BAOs with operators in F will be denoted by BAOF . The variety BAO{f}, where f is a unary operator, is usually referred to as the variety of modal algebras (the algebraic counterpart of modal logic). 1991 Mathematics Subject Classification: Primary 06E25; Secondary 03G15, 08A40, 03B45. The paper is in final form and no version of it will be published elsewhere.