191 research outputs found
A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions
Semi-Heyting algebras were introduced by the second-named author during
1983-85 as an abstraction of Heyting algebras. The first results on these
algebras, however, were published only in 2008 (see [San08]). Three years
later, in [San11], he initiated the investigations into the variety DHMSH of
dually hemimorphic semi-Heyting algebras obtained by expanding semi-Heyting
algebras with a dually hemimorphic operation. His investigations were continued
in a series of papers thereafter. He also had raised the problem of finding
logics corresponding to subvarieties of DHMSH, such as the variety DMSH of De
Morgan semi-Heyting algebras, and DPCSH of dually pseudocomplemented
semi-Heyting algebras, as well as logics to 2, 3, and 4-valued DHMSH-matrices.
In this paper, we first present a Hilbert-style axiomatization of a new
implicative logic called--Dually hemimorphic semi-Heyting logic, (DHMSH, for
short)-- as an expansion of semi-intuitionistic logic by the dual hemimorphism
as the negation and prove that it is complete with respect to the variety DHMSH
of dually hemimorphic semi-Heyting algebras as its equivalent algebraic
semantics (in the sense of Abstract Algebraic Logic). Secondly, we characterize
the (axiomatic) extensions of DHMSH in which the Deduction Theorem holds.
Thirdly, we present several logics, extending the logic DHMSH, corresponding to
several important subvarieties of the variety DHMSH, thus solving the problem
mentioned earlier. We also provide new axiomatizations for Moisil's logic and
the 3-valued Lukasiewicz logic.Comment: 66 pages, 3 figure
Representation of Nelson Algebras by Rough Sets Determined by Quasiorders
In this paper, we show that every quasiorder induces a Nelson algebra
such that the underlying rough set lattice is algebraic. We
note that is a three-valued {\L}ukasiewicz algebra if and only if
is an equivalence. Our main result says that if is a Nelson
algebra defined on an algebraic lattice, then there exists a set and a
quasiorder on such that .Comment: 16 page
Admissibility via Natural Dualities
It is shown that admissible clauses and quasi-identities of quasivarieties
generated by a single finite algebra, or equivalently, the quasiequational and
universal theories of their free algebras on countably infinitely many
generators, may be characterized using natural dualities. In particular,
axiomatizations are obtained for the admissible clauses and quasi-identities of
bounded distributive lattices, Stone algebras, Kleene algebras and lattices,
and De Morgan algebras and lattices.Comment: 22 pages; 3 figure
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