It is well-known that intuitionistic logics can be formalized by means of Heyting algebras, i.e. relatively pseudocomplemented semilattices. Within such algebras the logical connectives implication and conjunction are formalized as the relative pseudocomplement and the semilattice operation meet, respectively. If the Heyting algebra has a bottom element $0$, then the relative pseudocomplement with respect to $0$ is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with $0$ satisfying only the Ascending Chain Condition (these assumptions are trivially satisfied in finite meet-semilattices) and introduce the operators formalizing the connectives negation $x^{0}$ and implication $x\rightarrow y$ as the set of all maximal elements $z$ satisfying $x\wedge z=0$ and as the set of all maximal elements $z$ satisfying $x\wedge z\leq y$, respectively. Such a negation and implication is ‘unsharp’ since it assigns to one entry $x$ or to two entries $x$ and $y$ belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. We present several examples. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts.

中文翻译：

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代数结构将逻辑形式化，具有不明确的蕴涵和否定

众所周知，直觉逻辑可以通过Heyting代数（即相对伪补半格）形式化。在这样的代数中，逻辑连接词蕴涵和合取分别被形式化为相对伪补和半格运算的满足。如果 Heyting 代数有一个底元 $0$，则相对于 $0$ 的相对伪补称为伪补，在该逻辑中被视为连接否定。我们的想法是考虑一个任意的满足半格，$0$ 仅满足升链条件（这些假设在有限的满足半格中基本满足），并引入运算符形式化连接词否定 $x^{0}$ 和蕴涵 $x \rightarrow y$ 分别为满足 $x\wedge z=0$ 的所有最大元素 $z$ 的集合和满足 $x\wedge z\leq y$ 的所有最大元素 $z$ 的集合。这样的否定和蕴含是“不清晰的”，因为它将子集而不是半格子的元素分配给分别属于半格的一个条目$x$或两个条目$x$和$y$。令人惊讶的是，这种否定和蕴涵仍然具有直觉逻辑中这些连接词的许多性质，特别是派生规则Modus Ponens。此外，不清晰的否定和不清晰的蕴涵可以分别通过五个或七个简单公理来表征。我们举几个例子。介绍了演绎系统和过滤器的概念以及由这种过滤器确定的同余性。我们最后描述这些概念之间的某些关系。