基于k-循环模态伪补De Morgan代数的句子逻辑
Sentential logics based on k-cyclic modal pseudocomplemented De Morgan algebras
摘要：
模伪补De Morgan代数上算子理论的研究始于文[15]和[16]。本文引入并研究了一类被自同构k-周期（或称Ck-代数）充实的模态伪补De Morgan代数，其中k是正整数;当k = 2时，这类代数与文[15]中所研究的自同构k-周期（或称Ck-代数）是一个新的一元算子的情形相一致.首先证明了Ck-代数类是半单簇，并确定了生成代数。然后计算了生成元为n的自由Ck-代数的基数。在代数研究之后，我们建立了两个代数逻辑，它们的代数对应物是Ck-代数类，我们记为L<k和Lk，对于每一个k。Lk是一个1-断言逻辑，L<k是与Ck-代数类相关的保度逻辑。通过对这些逻辑的研究，我们证明了L<k是次协调的，它是原代数的，是代数等价的，但不能代数化。相反，我们证明了Lk是可代数化的，与L<k共享相同的定理，但不是次协调的。
英文摘要：
The study of the theory of operators over modal pseudocomplemented De Morgan algebras was begun in the papers [15] and [16]. In this paper, we introduce and study the class of modal pseudocomplemented De Morgan algebras enriched by an automorphism k-periodic (or Ck-algebras) where k is a positive integer; for k = 2 the class coincides with the one studied in [15] where the automorphism works as a new unary operator. In the first place, we prove the class Ck-algebras is a semisimple variety and we determine the generating algebras. Afterwards, we calculate the cardinality of the free Ck-algebra with n generator. After the algebraic study, we built two sentential logics that have as algebraic counterpart the class of Ck-algebras that we denote L<k and Lk for every k. Lk is a 1-assertional logic and L<k is the degree-preserving logic both associated with the class of Ck-algebras. Working over these logics, we prove that L<k is paraconsistent, which is protoalgebraic and finitely equivalential but not algebraizable. In contrast, we prove that Lk is algebraizable, sharing the same theorem with L<k, but not paraconsistent.
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