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Modalized Heyting calculus mHC introduced by Leo Esakia in his work, is the augmentation of the Intuitionistic logic Int by a modal operator . This modalized Heyting calculus is a weakening of the proof-intuitionistic logic KM of Kuznetsov and Muravitsky by discarding Löb

axiom. There is exact embedding of the mHC calculus into the modal system K4.Grz. Temporal Heyting calculus tHC is a temporal enrichement of mHC. This calculus was introduced by Leo Esakia. The temporal Heyting calculus tHC is defined on the basis of mHC with

additional axioms for the “adjoint” modality. Algebraic models of mHC are fHA-algebras (frontal Heyting algebras). In their work Jose Luis Castiglioni, Marta Sagastume, Hernan Javier San Martin have extended Heyting duality to the category fHA.

   We investigate variety of temporal Heyting algebras tHA, which represent algebraic models of temporal Heyting calculus tHC and have the following results:

1. We develop a theory of temporal Heyting algebras.

2. We generalize Heyting duality to the category tHA.

3. Characterization of simple tHA-algebras and subdirectly irreducible tHA-algebras is given. 

Modalized Heyting calculus mHC, introduced by Leo Esakia, is the augmentation of the intuitionistic logic Int by a modal operator ƒ. This modalized Heyting calculus is a weakening of the proof-intuitionistic logicKM of Kuznetsov and Muravitsky by discarding Lob’s axiom. There is an exact embedding of mHC into the modal system K4.Grz. Temporal Heyting calculus tHC is a temporal enrichement of mHC. This calculus was also introduced by Leo Esakia.

We investigate the variety of temporal Heyting algebras tHA, which represent algebraic models of temporal Heyting calculus tHC.We have the following results:

• We develop a theory of temporal Heyting algebras.

• We generalize Heyting duality to the category tHA.

• We characterize subdirectly irreducible and simple tHA-algebras.

Modalized Heyting calculus mHC, introduced by Leo Esakia, is the augmentation of the intuitionistic logic Int by a modal operator ƒ. This modalized Heyting calculus is a weakening of the proof-intuitionistic logicKM of Kuznetsov and Muravitsky by discarding Lob’s axiom. There is an exact embedding of mHC into the modal system K4.Grz. Temporal Heyting calculus tHC is a temporal enrichement of mHC. This calculus was also introduced by Leo Esakia.

We investigate the variety of temporal Heyting algebras tHA, which represent algebraic models of temporal Heyting calculus tHC.We have the following results:

• We develop a theory of temporal Heyting algebras.

• We generalize Heyting duality to the category tHA.

• We characterize subdirectly irreducible and simple tHA-algebras.

A ”symmetric” formulation of intuitionistic propositional calculus Int² , suggested by various authors (G. Moisil, A. Kuznetsov, C. Rauszer), presupposes that each of the connectives &, ∨, ®, T, ⊥ has its dual ∨, &, ®, ⊥, T, and the duality principle of the classical logic is restored. G¨odel logic is the extension of intuitionistic logic by linearity axiom: (p → q)∨ (q → p). Denote by Gn the n valued G¨odel logic. We investigate symmetric G¨odel logic G²n , the language of which is enriched by two modalities ƒ₁,ƒ₂. The resulting system is named bimodal symmetric G¨odel logic and is denoted by MG²n . MG²n -algebras represent algebraic models of the logic MG²n. The variety MG²n of all MG²n -algebras is generated by finite linearly ordered MG² -algebras of finite height m, where 1 ≤ m ≤ n. We focus on MG²n algebras, which correspond to n valued MG² n logic. A description and characterization of m-generated free and projective MG² - algebras in the variety MG²n is given.

A ”symmetric” formulation of intuitionistic propositional calculus Int² , suggested by various authors (G. Moisil, A. Kuznetsov, C. Rauszer), presupposes that each of the connectives &, ∨, ®, T, ⊥ has its dual ∨, &, ®, ⊥, T, and the duality principle of the classical logic is restored. G¨odel logic is the extension of intuitionistic logic by linearity axiom: (p → q)∨ (q → p). Denote by Gn the n valued G¨odel logic. We investigate symmetric G¨odel logic G²n , the language of which is enriched by two modalities ƒ1,ƒ2. The resulting system is named bimodal symmetric G¨odel logic and is denoted by MG²n . MG²n -algebras represent algebraic models of the logic MG²n. The variety MG²n of all MG²n -algebras is generated by finite linearly ordered MG² -algebras of finite height m, where 1 ≤ m ≤ n. We focus on MG²n algebras, which correspond to n valued MG² n logic. A description and characterization of m-generated free and projective MG² - algebras in the variety MG²n is given.

A ”symmetric” formulation of intuitionistic propositional calculus Int² , suggested by various authors (G. Moisil, A. Kuznetsov, C. Rauszer), presupposes that each of the connectives &, ∨, ®, T, ⊥ has its dual ∨, &, ®, ⊥, T, and the duality principle of the classical logic is restored. G¨odel logic is the extension of intuitionistic logic by linearity axiom: (p → q)∨ (q → p). Denote by Gn the n valued G¨odel logic. We investigate symmetric G¨odel logic G²n , the language of which is enriched by two modalities ƒ₁,ƒ₂. The resulting system is named bimodal symmetric G¨odel logic and is denoted by MG²n . MG²n -algebras represent algebraic models of the logic MG²n. The variety MG²n of all MG²n -algebras is generated by finite linearly ordered MG² -algebras of finite height m, where 1 ≤ m ≤ n. We focus on MG²n algebras, which correspond to n valued MG² n logic. A description and characterization of m-generated free and projective MG² - algebras in the variety MG²n is given.

In this work we deal with algebraic counterparts of Lukasiewicz logic enriched by finite number of truth constants. A propositional many-valued logical system which turned out to be equivalent to the expansion of Lukasiewicz Logic L by adding into the language a truth-constant r for each real r (0, 1), together with a number of additional axioms was proposed by Pavelka. We investigate the varieties of algebras, corresponding to of Lukasiewicz logic enriched by finite number of truth constants. Specifically we show that these varieties contain non-trivial minimal subvariety generated by finite linearly ordered algebra which is functionally equivalent to Post algebra. The analysis and charachterizations of appropriate varieties and corresponding logical systems are given. Free and Projective algebras are studied in these varieties as well as projective formulas and unification problem.

A ”symmetric” formulation of intuitionistic propositional calculus Int² , suggested by various authors (G. Moisil, A. Kuznetsov, C. Rauszer), presupposes that each of the connectives &, ∨, ®, T, ⊥ has its dual ∨, &, ®, ⊥, T, and the duality principle of the classical logic is restored. G¨odel logic is the extension of intuitionistic logic by linearity axiom: (p → q)∨ (q → p). Denote by Gn the n valued G¨odel logic. We investigate symmetric G¨odel logic G²n , the language of which is enriched by two modalities ƒ₁,ƒ₂. The resulting system is named bimodal symmetric G¨odel logic and is denoted by MG²n . MG²n -algebras represent algebraic models of the logic MG²n. The variety MG²n of all MG²n -algebras is generated by finite linearly ordered MG² -algebras of finite height m, where 1 ≤ m ≤ n. We focus on MG²n algebras, which correspond to n valued MG² n logic. A description and characterization of m-generated free and projective MG² - algebras in the variety MG²n is given.

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