Multiple Conclusion Rules in Logics with the Disjunction Property

We prove that for the intermediate logics with the disjunction property any basis of admissible rules can be reduced to a basis of admissible m-rules (multiple-conclusion rules), and every basis of admissible m-rules can be reduced to a basis of admissible rules. These results can be generalized to a broad class of logics including positive logic and its extensions, Johansson logic, normal extensions of S4, n-transitive logics and intuitionistic modal logics

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

Singly generated quasivarieties and residuated structures

A quasivariety K of algebras has the joint embedding property (JEP) iff it is generated by a single algebra A. It is structurally complete iff the free countably generated algebra in K can serve as A. A consequence of this demand, called "passive structural completeness" (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if K is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if K is relatively semisimple then it is generated by one K-simple algebra. It is a minimal quasivariety if, moreover, it is PSC but fails to unify some finite set of equations. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC iff its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics