Revaz Grigolia

MV -algebras are the algebraic counterpart of the infinite valued Łukasiewicz sentential calculus, as Boolean algebras are with respect to the classical propositional logic. As it is well known, MV - algebras form a category which is equivalent to the category of abelian lattice ordered groups (ℓ-groups, for short) with strong unit. It is known that any subvariety of MV -algebras is generated by finitely many algebras. Notice that the free algebras over the subvarieties of MV -algebras have been described functionally. Finitely generated free MVn-algebras (that correspond to n-valued Lukasiewicz logic) was described algebraically. In this paper we give an algebraic description of finitely many generated free MV -algebras in the variety V(S₁^ω) generated by the algebra S₁^ω (denoted as C) that was introduced by Chang and later by Komori. Moreover, we give ordered spectral spaces of the free algebras. In this work it is represented the free algebras by means of subdirect product of the finite family of chains. 

It is introduced a new algebra (A, ⊗,⊕,∗,⇀, 0, 1) called LPG-algebra if (A, ⊗,⊕,∗,0, 1) is LP - algebra (i.e. an algebra from the variety generated by perfect MV -algebras) and (A,⇀, 0, 1) is a Gödel algebra (i.e. Heyting algebra satisfying the identity (x ⇀ y) ∨ (y ⇀ x) = 1). The lattice of congruences of an LPG-algebra (A, ⊗, ⊕, ∗,⇀, 0, 1) is isomorphic to the lattice of Skolem filters (i.e. special type of MV -filters) of the MV -algebra (A,⊗,⊕,∗,0, 1). The variety LPG of LPG-algebras is generated by the algebras (C, ⊗, ⊕, ∗,⇀, 0, 1) where (C, ⊗, ⊕, ∗, 0, 1) is Chang MV -algebra. Any LPG-algebra is bi-Heyting algebra. The set of theorems of the logic LPG is recursively enumerable. Moreover, the description of finitely generated free LPG-algebras and their dual objects – MV -spaces is given.

MV-algebras (with signature (⊕,⊗,⇀, 0, 1)) and type (2,2,1,0,0)) are the algebraiccounterpart of the infinite valued Lukasiewicz sentential calculus [1], as Boolean algebras are with respect to the classical propositional logic. In contrast to what happens for Boolean algebras, there are MV-algebras which are not semi-simple, i.e. the intersection of their maximal ideals (the radical of A) is different from {0}. Non-zero elements from the radical of A are called infinitesimals and denoted by Rad(A). 

The MV-algebra A is called perfect if A = R* (A) = Rad(A) is the intersection of all maximal filters of A. The class of perfect MV -algebras does not form a variety and contains non-simple subdirectly irreducible MV-algebras. The variety V (S₁^w) is generated by all perfect MV-algebras, it is also generated by a single MV-chain S₁^w (=C ) (that is perfect) defined by Chang. We name by S1w -algebras all the algebras from the variety, generated by S1w .

S₁^w(0) =Γ(Z, 1), S1w(1) = S₁^w =C=Γ(Z×lexZ, (1, 0)) with generator (0, 1) , S1w(1)= Γ(Z× lexZ×lexZ,(1, 0, 0) with generators (0, 0, 1), (0, 1, 0), where Γ is a well known Mundici’s functor. from the category of the lattice ordered group with a strong unit to the category of MV-algebras, the number of factors Z is equal to 2+1, and ⨯lex is the lexicographic product.

It is shown that the lattice of congruences of a GMV -algebra (A,⊗,⊕,∗,⇀,0,1)(A,⊗,⊕,∗,⇀,0,1) is isomorphic to the lattice of Skolem filters (i. e. special type of MV-filters) of the MV-algebra (A,⊗,⊕,∗,0,1)(A,⊗,⊕,∗,0,1). Any GMV-algebra is bi-Heyting algebra. Any chain GMV-algebra is simple, and any GMV-algebra is semi-simple. Finitely generated GMV-algebras are described, and finitely generated finitely presented GMV-algebras are characterized. The algebraic counterpart of axiomatically presented GMV-logic is GMV-algebras.

Following K. Segerberg and V. Pratt, who have been introduced dynamic propositional logic [2] and dynamic algebras [1], dynamic propositional Lukasiewicz logic DPL and dynamic MV-algebras are introduced and theories of the logic DPL and dynamic MV-algebras are developed. Dynamic MV-algebras are algebraic counterparts of the logic DPL, that in turn represent two-sorted algebras that combine the ) resemblingàvarieties of MV-algebras and regular algebras into a single finitely axiomatized variety (M, R, . Kripke semantics is developed for dynamic propositionalàR-module with "scalar" multiplication Lukasiewicz logic.

An algebra, (A, ⊕, ⊗, *, Ù, Ú, ® 0, 1) is called LPG-algebra if (A, ⊕, ⊗,*, 0, 1) is LP algebra (i. e. an algebra from the variety generated by perfect MV-algebras and (A, Ù, Ú,→0, 1) is a Gödel algebra (i. e. Heyting algebra satisfying the identity (x ®y) Ú (y®x)=1).

Example. The algebra  (C, ⊕, ⊗, *, Ù,Ú, ® 0, 1) where (C, ⊕, ⊗, *, 0, 1) is Chang MV algebra, is LPG -algebra.

We construct a covariant functor γγ from the category of monadic MV-algebras into the category of Q-distributive lattices, i.e., distributive lattices with quantifier introduced by R. Cignoli. For every monadic MV-algebra, we construct a dual object named QM-space; these objects form a special subcategory of spectral spaces and of Q-spaces developed by R. Cignoli for Q-distributive lattices.

The predicate Lukasiewicz (infinitely valued) logic QL is defined in the following standard way. The existential (universal) quantfier is interpreted as supremum (infimum) in a complete MV –algebra. Then the valid formulas of predicate calculus are defined as all formulas having value 1 for any assignment. The functional description of the predicate calculus is given by Rutledge. Scarpellini has proved that the set of valid formulas is not recursively enumerable.

We introduce the new logic CL, which is an extension of the infinitely valued Lukasiewicz logic L, the language of which enriched by 0-ary connective c that is interpreted as ”quasi-false”, the algebraic counterpart of which are algebras from a quasi-variety generated by the perfect Chang MV -algebras C signature of which enriched by the constant element c. For this aim we introduce a new class CL of algebras which is a quasi-variety and the algebras from this quasi-variety we name CL-algebras. Adding a new inference rule to the logic CL, thereby increased a deducibility power, we introduce the logic CL+ and defining the notion of quasi-true (q-true) formulas it is proved the completeness theorem for this logic.

The theory of three-valued multi modal epistemic Łukasievicz logic with constant, which is an extension of the three-valued Łukasiewicz logic, the language of which is extended by nullary and unary connectives is developed in this paper. The unary connectives are interpreted as modal operators (knowledge operators). We propose to use such logic in studying immune system. A relational system is developed as a semantic of this logic. The relational systems represent the immune system which in its turn is a part of relational biology. 

A description of finitely generated free monadic MV-algebras and a characterization of projective monadic MV-algebras in locally finite varieties is given. It is shown that unification type of locally finite varieties is unitary.

A characterization of finitely generated projective MV-algebras is given, introducing the new notions of correct partitions and e-relatively complete subalgebras

The variety of MV (C)-algebras generated by perfect MV -algebras and the logic of this variety is investigated. Perfect MV (C)-algebras are those MV -algebras generated by their infinitesimal elements or, equivalently, generated by their radical, where radical is the intersection of all maximal ideals. Dual objects of these algebras are constructed. It is shown that finitely generated MV (C)-algebra is projective if it is finitely presented. Unification type of the variety of MV (C)-algebras is 1. Moreover, the variety of MV (C)-algebras is structurally complete.

The class of MV-algebras arises as the algebraic counterpart of the infinite valued Lukasiewciz sentential calculus, as Boolean algebras did with respect to the classical propositional logic. Perfect MV-algebras are those MV-algebras generated by their infinitesimal elements or, equivalently, generated by their radical. Hence perfect MV-algebras can be seen as extreme examples of non-archimedean MV-algebras. We study the problem of unification and structural completeness of the logic LP corresponding to the perfect MV-algebras. For this aim the duality theory for the variety of algebras generated by perfect MV-algebras is developed.

A characterization of finitely generated Heyting algebras and Boolean algebras with operators such as Closure algebras, Grzegorchik algebras, diagonalizable algebras. A description of finitely generated projective and free algebras in these varieties of algebras is given.

A description of the free cyclic algebra over the variety of Solovay algebras, as well as over its pyramid locally finite subvarieties is given.

We describe the class of finitely generated projective subalgebras of the finitely generated free MV -algebras. As main results we characterize such algebras by means a functional description of their generators, obtaining in such a way, up to isomorphism, a characterization of all finitely generated projective MV -algebras. Also we get a characterization of finitely generated projective MV -algebras via special formulas, the projective formulas, of Lukasiewicz logic.

The variety MG2 of bimodal symmetric Gödel algebras, which represent the algebraic counterparts of bimodal symmetric Gödel logic MG2 , is investigated. Description of free algebras and characterization of projective bimodal symmetric MG2 -algebras is given.

A new logic - Lukasiewicz logic enriched with constant connective, and corresponding to it variety of algebras, is introduced. The unification problems are analyzed for the new logic. It is shown that the logic has finitary unification type.

The unification problems are analyzed for bimodal (temporal) logical system LinTGrz. It is shown that the logic has finitary unification type.

We define and study monadic BL-algebras as a pair of BL-algebras one of which is a special case of relatively complete subalgebra.

A characterization of finitely generated projective MV-algebras is given, introducing the new notions of correct partitions and e-relatively complete subalgebras.

Co-product of Heyting algebras

The existence of a unique structurally complete intermediate logic having the disjunction property is proved. And this unique intermediate logic is well-known Medvedev’s logic.

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