论文标题
常规双$ p $ - 代数：与katriňák的定理和申请
Regular Double $p$-Algebras: A converse to a Katriňák's Theorem, and Applications
论文作者
论文摘要
1973年，卡特里王（Katriéák）证明，常规的双$ p $ - 代数可以被视为（常规的）双重嘿代数，通过巧妙地构建二进制术语，以实现heying的含义及其双重二元，并在伪造及其双重方面。在本文中，我们证明了与Katriňák的定理相反，从某种意义上说，在常规双重伪造的Heyting代数的各种rdpch中，暗示操作$ \ to $ to $ to com of to $ \ $ to com of to $ \ of to $ \ of to $ \ $。 As applications of this result together with the above-mentioned Katriňák's theorem, we show that the varieties RDBLP, RDPCH, RPCH$^d$ and RDBLH of regular double $p$-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are既等同于彼此的术语，也认为品种rdmp，rdmh，rdmdblh，rdmdblh，常规de Morgan $ p $ -p $ -algebras，常规的de Morgan Heyting代数，常规的De Morgan Double Heyting代数和常规的De Morgan de Morgan de Morgan de Morgan de de Morgan Double $ P $ - Algebras，也分别与对方相同。从这些结果和亚当斯，Sankappanavar和Vaz de Carvalho的最新结果中，我们推断出所有这些品种的子变量的晶格具有基数$ 2^{\ Aleph_0} $。然后，我们定义了新的逻辑，RDPCH，RPCHD和RDMH，并表明它们与RDPCH，RPCH $^d $和RDMH分别为代数，作为其等效的代数语义。还推断出所有上述逻辑的扩展的晶格具有基数$ 2^{\ aleph_0} $。
In 1973, Katriňák proved that regular double $p$-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heying implication and its dual in terms of pseudocomplement and its dual. In this paper we prove a converse to the Katriňák's theorem, in the sense that in the variety RDPCH of regular dually pseudocomplemented Heyting algebras, the implication operation $\to$ satisfies the Katriňák's formula. As applications of this result together with the above-mentioned Katriňák's theorem, we show that the varieties RDBLP, RDPCH, RPCH$^d$ and RDBLH of regular double $p$-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties RDMP, RDMH, RDMDBLH, RDMDBLP of regular De Morgan $p$-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double $p$-algebras, respectively, are also term equivalent to each other. From these results and recent results of Adams, Sankappanavar and vaz de Carvalho, we deduce that the lattices of subvarieties of all these varieties have cardinality $2^{\aleph_0}$. We then define new logics, RDPCH, RPCHd, and RDMH, and show that they are algebraizable with RDPCH, RPCH$^d$ and RDMH, respectively as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality $2^{\aleph_0}$.
