关于有界半关联半格和半关联格的Kalman函子
On Kalman's functor for bounded hemi-implicative semilattices and hemi-implicative lattices
摘要：
半蕴涵半格（格）最初是在弱蕴涵半格（格）的名义下定义的。半蕴涵半格是一个$(2,2,0)$型的代数$(H,\wedge,\rightarrow,1)$，使得$(H,\wedge)$是交半格，$1$是关于阶的最大元素，$a\rightarrow a = 1$对于每个$a\in H$和每个$a$，$b$，$c\in H$，如果$a\leq b\rightarrow c$则$a\wedge b \leq c$。一个有界半蕴涵半格是一个$(2,2,0,0)$型的代数$(H,\wedge,\rightarrow,0,1)$，使得$(H,\wedge,\rightarrow,1)$是一个半蕴涵半格，并且$0$是关于阶的第一个元素。半关联格是$(2,2,2,0,0)$型的代数$(H,\wedge,\vee,\rightarrow,0,1)$，使得$(H,\wedge,\vee,0,1)$是有界分配格，并且约简代数$(H,\wedge,\rightarrow,1)$是半关联半格。在本文中，我们引入了一个等价的范畴有界半蕴涵半格和半蕴涵格，分别，这是由一个旧的建设，由于J.Kalman的有界分配格和Kleene代数。
英文摘要：
Hemi-implicative semilattices (lattices), originally defined under the name of weak implicative semilattices (lattices), were introduced by the second author of the present paper. A hemi-implicative semilattice is an algebra $(H,\wedge,\rightarrow,1)$ of type $(2,2,0)$ such that $(H,\wedge)$ is a meet semilattice, $1$ is the greatest element with respect to the order, $a\rightarrow a = 1$ for every $a\in H$ and for every $a$, $b$, $c\in H$, if $a\leq b\rightarrow c$ then $a\wedge b \leq c$. A bounded hemi-implicative semilattice is an algebra $(H,\wedge,\rightarrow,0,1)$ of type $(2,2,0,0)$ such that $(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice and $0$ is the first element with respect to the order. A hemi-implicative lattice is an algebra $(H,\wedge,\vee,\rightarrow,0,1)$ of type $(2,2,2,0,0)$ such that $(H,\wedge,\vee,0,1)$ is a bounded distributive lattice and the reduct algebra $(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice. In this paper we introduce an equivalence for the categories of bounded hemi-implicative semilattices and hemi-implicative lattices, respectively, which is motivated by an old construction due J. Kalman that relates bounded distributive lattices and Kleene algebras.
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