Positive Modal Hilbert Algebras and Fischer Servi Modal Hilbert Algebras

The concept of soft sets, introduced by Molodtsov (20) is a math- ematical tool for dealing with uncertainties, that is free from the diculties that have troubled the traditional theoretical approaches. In this paper, we apply the notion of the soft sets of Molodtsov to the theory of Hilbert algebras. The notion of soft Hilbert (abysmal and deductive) algebras, soft subalgebras, soft abysms and soft deductive systems are introduced, and their basic proper- ties are investigated. The relations between soft Hilbert algebras, soft Hilbert abysmal algebras and soft Hilbert deductive algebras are also derived.

This paper introduces the concepts of [Formula: see text]-Implication algebras (Hilbert algebras, KU algebras, JU algebras, BE algebras, and CI algebras) each serving as a generalization of their respective counterparts: Implication algebras, Hilbert algebras, KU-algebras, BE-algebras, and CI-algebras. It is established that every commutative, self-distributive [Formula: see text]-BE algebra is a [Formula: see text]-Hilbert algebra, and that any [Formula: see text]-Implication algebra is [Formula: see text]-BE algebras. Furthermore, every [Formula: see text]-Hilbert algebra is shown to be a self-distributive [Formula: see text]-BE algebra. We prove that [Formula: see text]-Implication algebras are [Formula: see text]-Hilbert algebras(KU-algebras, BE-algebras). We study some properties of [Formula: see text]-Implication algebras (Hilbert algebras, KU-algebras, JU-algebras, BE algebras, and CI algebras). We establish that every commutative and self-distributive [Formula: see text]-BE algebra is a [Formula: see text]-Hilbert algebra.

In this paper, we introduce the concept of roughness in the context of Hilbert algebras, a class of algebraic structures fundamental to studying non-classical logic. By integrating rough set theory with Hilbert algebras, we investigate the lower and upper approximations of subalgebras and ideals. We show that the lower and upper approximations of a subalgebra (or ideal) in a Hilbert algebra also make up a subalgebra (or ideal). This implies that algebraic systems can employ rough set concepts. Our results demonstrate that the approximation spaces induced by ideals in Hilbert algebras provide a robust framework for analyzing algebraic structures under incomplete or uncertain information. Furthermore, we present illustrative examples to validate our theoretical findings and highlight the practical implications of this approach. This study not only enriches the theoretical foundations of rough set theory but also opens new avenues for its application in algebraic logic and related fields.

In this paper, we will study some connections between Hilbert algebras and binary block-codes. With these codes, we can eassy obtain orders which determine supplementary properties on these algebras. We will try to emphasize how, using binary block-codes, we can provide examples of classes of Hilbert algebras with some properties, in our case, classes of semisimple Hilbert algebras and classes of local Hilbert algebras.

This work introduces the concept of f-derivations in Hilbert algebras, exploring its theoretical foundation alongside a series of illustrative examples. We examine fundamental properties associated with f-derivations through rigorous analysis, shedding light on their algebraic structure and behaviour. In particular, we demonstrate that the kernel Kerdf(A) constitutes a near filter (subalgebra), while the fixed set Fixdf(f) forms a subalgebra within the Hilbert algebra A. These results provide new insights into the interaction between derivations and substructures in Hilbert algebras, offering potential avenues for further exploration in algebraic logic and related fields.

A Hilbert algebra with supremum is a Hilbert algebra where the associated order is a join-semilattice. This class of algebras is a variety and was studied in Celani and Montangie (2012). In this paper we shall introduce and study the variety of $${H_{\Diamond}^{\vee}}$$H??-algebras, which are Hilbert algebras with supremum endowed with a modal operator $${\Diamond}$$?. We give a topological representation for these algebras using the topological spectral-like representation for Hilbert algebras with supremum given in Celani and Montangie (2012). We will consider some particular varieties of $${H_{\Diamond}^{\vee}}$$H??-algebras. These varieties are the algebraic counterpart of extensions of the implicative fragment of the intuitionistic modal logic $${\mathbf{IntK}_{\Diamond}}$$IntK?. We also determine the congruences of $${H_{\Diamond}^{\vee}}$$H??-algebras in terms of certain closed subsets of the associated space, and in terms of a particular class of deductive systems. These results enable us to characterize the simple and subdirectly irreducible $${H_{\Diamond}^{\vee }}$$H??-algebras.

In this paper, we will study the class of Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semilattice. First, we will give a simplified topological duality for Hilbert algebras using sober topological spaces with a basis of open-compact sets satisfying an additional condition. Next, we will extend this duality to Hilbert algebras with supremum. We shall prove that the ordered set of all ideals of a Hilbert algebra with supremum has a lattice structure. We will also see that in this lattice, it is possible to define an implication, but the resulting structure is neither a Heyting algebra nor an implicative semilattice. Finally, we will give a dual description of the lattice of ideals of a Hilbert algebra with supremum.

In this paper, we will study a particular subvariety of Hilbert algebras with a modal operator $$\square $$, called Lax Hilbert algebras. These algebras are the algebraic semantic of the $$\left\{ \square ,\rightarrow \right\} $$-fragment of a particular intuitionistic modal logic, called Propositional Lax Logic ($$\mathcal {PLL}$$), which has applications to the formal verification of computer hardware. These algebras turn to be a generalization of the variety of Heyting algebras with a modal operator studied, under different names, by Macnab (Algebra Univ 12:5–29, 1981), Goldblatt (Math Logic Q 27(31–35):495–529, 1981; J Logic Comput 21(6):1035–1063, 2010) and by Bezhanishvili and Ghilardi (Ann Pure Appl Logic 147:84–100, 2007). We shall prove that the set of fixpoints of a Lax Hilbert algebra $$\left\langle A,\square \right\rangle $$ is a Hilbert algebra such that its dual space is homeomorphic to the subspace of reflexive elements of the dual space of A. We will define the notion of subframe of a Hilbert space $$\left\langle X,{\mathcal {K}}\right\rangle $$, and we will prove that there is a 1–1 correspondence between subframes of $$\left\langle X,{\mathcal {K}}\right\rangle $$ and binary relations $$Q\subseteq X\times X$$ such that $$\left\langle X,{\mathcal {K}},Q\right\rangle $$ is a Lax Hilbert space. In addition, we will define the notion of subframe variety and we will prove that any variety of Hilbert algebras is a subframe variety.

In this paper, we introduce the notion of implication filter as a generalization of Boolean filter of first kind in Hilbert algebra and study it in detail. Also, we prove that F is an implication filter of a Hilbert algebra H if and only if every implicative filter of quotient algebra H / F is an implication filter. Finally, we generalized Boolean algebra and introduced a weak implication algebra, we prove that F is an implication filter of a Hilbert algebra H if and only if the quotient algebra H / F is a weak implication algebra. By suitable diagrams, we summarize the results of this paper and the previous results in these fields.

Hilbert algebra with a Hilbert-Galois connection, or HilGC-algebra, is a triple \(\left(A,f,g\right)\) where \(A\) is a Hilbert algebra, and \(f\) and \(g\) are unary maps on \(A\) such that \(f(a)\leq b\) iff \(a\leq g(b)\), and \(g(a\rightarrow b)\leq g(a)\rightarrow g(b)\) forall \(a,b\in A\). In this paper, we are going to prove that some varieties of HilGC-algebras are characterized by first-order conditions defined in the dual space and that these varieties are canonical. Additionally, we will also study and characterize the congruences of an HilGC-algebra through specific closed subsets of the dual space. This characterization will be applied to determine the simple algebras and subdirectly irreducible HilGC-algebras.

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.

In this paper, we introduce a Sheffer stroke Hilbert algebra by giving definitions of Sheffer stroke and a Hilbert algebra. After it is shown that the axioms of Sheffer stroke Hilbert algebra are independent, it is given some properties of this algebraic structure. Then it is stated the relationship between Sheffer stroke Hilbert algebra and Hilbert algebra by defining a unary operation on Sheffer stroke Hilbert algebra. Also, it is presented deductive system and ideal of this algebraic structure. It is defined an ideal generated by a subset of a Sheffer stroke Hilbert algebra, and it is constructed a new ideal of this algebra by adding an element of this algebra to its ideal.

In this paper, we first introduce the notion of Stonean implicative filter in Hilbert algebra and study it in detail. Finally, we introduce a Stonean Hilbert algebra and investigate its properties, we prove that, H is a Stonean Hilbert algebra if and only if the set of all regular elements of H is a Stonean Hilbert algebra. By the diagrams, we summarize the results of this paper and give the relationships between all types of filters in a Hilbert algebra. Also, we give the relationships between Hilbert algebra and some of algebraic structures.

The concept of deductive system on a Hilbert algebra was introduced by A. Diego. We show that the set Ded A of all deductive systems on a Hilbert algebra A forms an algebraic lattice which is distributive.

The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra i ?both algebras have the same ?lters.An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice),and conversely.The implication in an implicative partial semilattice is characterised in terms of ?lters of the underlying partial semilattice.