On the structure of varieties with equationally definable principal congruences IV
On the structure of varieties with equationally definable principal congruences IV
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论等式可定义主同余的簇结构 IV
DOI：
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
D. Pigozzi
中科院分区：
文献类型：
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作者：
W. Blok;D. Pigozzi
引入了拓扑-内部代数的概念，它是一个（拓扑）内部代数和剩余偏序幺半群的杂交。伪内部代数的初等算术的发展导致一个简单的等式公理化。研究了类似于内部代数的开滤子的开滤子概念。伪内部代数以代数形式表示具有可交换的、规则的三元演绎（TD）项p（x，y，z）的簇中固有的逻辑，其由以下条件定义：（1）p（x，y，z）≡ z（modΘ（x，y））：（2）对代数A的固定元素a，B，{p（a，B，z）：z ∈ A}是Θ（a，B）的等价类集的横截;（3）当θ（a，B）=Θ（a ′，B′）时，p（a，B，z）和p（a ′，B ′，z）定义相同的断面;（4）对于某一常数项1，Θ（p（x，y，1），1）= Θ（x，y）. TD项推广了（仿射）三元矩阵。具有交换的正则TD项的变种包括大多数传统代数逻辑的变种以及所有双点仿射群变种和n-幂环（剩余交换po-monoid，其中偏序是逆整除的）。主要定理：一个簇有一个可交换的，正则的TD项当且仅当它在项上定义等价于一个伪内部代数，其附加运算以一种自然的方式与开滤子相容。
The notion of apseudo-interior algebra is introduced; it is a hybrid of a (topological) interior algebra and a residuated partially ordered monoid. The elementary arithmetic of pseudo-interior algebras is developed leading to a simple equational axiomatization. A notion ofopen filter analogous to the open filters of interior algebras is investigated. Pseudo-interior algebras represent, in algebraic form, the logic inherent in varieties with acommutative, regular ternary deductive (TD) term p(x, y, z), which is defined by the conditions: (1)p(x,y,z) ≡ z (modΘ(x, y)); (2) for fixed elementsa, b of an algebra A, {p(a, b, z):z ∈ A} is a transversal of the set of equivalence classes of Θ(a, b); (3)p(a, b, z) andp(a′,b′,z) define the same transversal wheneverΘ(a,b)=Θ(a′,b′); (4)Θ(p(x, y, 1), 1)= Θ(x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator. Varieties with a commutative, regular TD term include most of the varieties of traditional algebraic logic as well as all double-pointed affine discriminator varieties andn-potent hoops (residuated commutative po-monoids in which the partial ordering is inverse divisibility). The main theorem:A variety has a commutative, regular TD term iff it is termwise definitionally equivalent to a pseudo-interior algebra with additional operations that are compatible with the open filters in a natural way.