正则伪补de Morgan代数簇
Varieties of Regular Pseudocomplemented de Morgan Algebras
摘要：
本文研究了值域为$n$的正则伪补de Morgan代数和Kleene代数的簇$\mathbf M_n$和$\mathbf K_n$. Priestley对偶，因为它适用于伪补德摩根代数使用。我们在$\mathbf M_n$中构造了单（等价地，亚直不可约）代数的对偶空间，并在$\mathbf M_1$和$\mathbf K_1$中明确地描述了单代数的对偶空间.我们证明了簇$\mathbf M_1$是局部有限的，但这个性质并不扩展到$\mathbf M_n$，甚至$n \geq 2$的$\mathbf K_n$。我们还证明了$\mathbf K_1$的子簇的格是一个$ω+ 1$链，并且$\mathbf K_2$或$\mathbf M_1$的子簇的格的基数是$2^ω$。给出了$\mathbf M_1$的子簇的格的描述。
英文摘要：
In this paper, we investigate the varieties $\mathbf M_n$ and $\mathbf K_n$ of regular pseudocomplemented de Morgan and Kleene algebras of range $n$, respectively. Priestley duality as it applies to pseudocomplemented de Morgan algebras is used. We characterise the dual spaces of the simple (equivalently, subdirectly irreducible) algebras in $\mathbf M_n$ and explicitly describe the dual spaces of the simple algebras in $\mathbf M_1$ and $\mathbf K_1$. We show that the variety $\mathbf M_1$ is locally finite, but this property does not extend to $\mathbf M_n$ or even $\mathbf K_n$ for $n \geq 2$. We also show that the lattice of subvarieties of $\mathbf K_1$ is an $ω+ 1$ chain and the cardinality of the lattice of subvarieties of either $\mathbf K_2$ or $\mathbf M_1$ is $2^ω$. A description of the lattice of subvarieties of $\mathbf M_1$ is given.
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