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Nearlattices. (English) Zbl 1151.06004

To describe algebraic properties of the connective of implication in logics, we use the so-called implication algebras. It turns out that the corresponding algebraic structures are join-semilattices where each principal filter is a lattice with respect to the induced order and, moreover, they are equipped with an antitone involution which is a complementation. However, this approach cannot be generalized for implication in intuitionistic logic because the connective of implication is identified with the relative pseudocomplementation which is not an involution; moreover, in intuitionistic logic the connectives of conjunction, disjunction and negation are independent. This motivates the authors to use a quite general approach: they treat join-semilattices where each principal filter is a lattice and, when axiomatizing them, they introduce a section pseudocomplementation (similarly as for lattices) to reach another kind of intuitionistic logic. Hence, they introduce the notion of a nearlattice, that is, a join-semilattice where the principal filter is a lattice with respect to the induced order. Also, a nearlattice can be described as an algebra with one ternary operation satisfying eight simple identities, hence, the class of nearlattices is a variety (a variety which is congruence distributive).
The authors define the notions of distributive and dually distributive algebra of type (3) and show that these two notions are equivalent for nearlattices (also equivalent with the fact that in the associated semilattices every principal filter is a distributive lattice), hence the variety of distributive nearlattices is locally finite. Finally, a nearlattice whose principal filters are pseudocomplemented lattices is called a sectionally pseudocomplemented nearlattice. The authors prove that the class of all sectionally pseudocomplemented nearlattices is a variety and every finite distributive nearlattice is sectionally pseudocomplemented.

MSC:

06A12 Semilattices
03G10 Logical aspects of lattices and related structures
06B20 Varieties of lattices
03B20 Subsystems of classical logic (including intuitionistic logic)
08B10 Congruence modularity, congruence distributivity
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References:

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