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We introduce the notion of distributivity for implicative-orthomodular lattices, proving an analogue result of the Foulis-Holland theorem. Based on this result, we characterize the distributive implicative-orthomodular lattices. Moreover, we define the center of an implicative-orthomodular lattice as the set of all elements that commute with all other elements, and we prove that the center is an implicative-Boolean algebra. Additionally, we give new characterizations of implicative-orthomodular lattices.

Based on implicative involutive BE algebras, we redefine the orthomodular lattices, by introducing the notion of implicative-orthomodular lattices, and we study their properties. We characterize these algebras, proving that the implicative-orthomodular lattices are quantum-Wajsberg algebras. We also define and characterize the implicative-modular algebras as a subclass of implicative-orthomodular lattices. The orthomodular softlattices and orthomodular widelattices are also redefined, by introducing the notions of implicative-orthomodular softlattices and implicative-orthomodular widelattices. Finally, we prove that the implicative-orthomodular softlattices are equivalent to implicative-orthomodular lattices and that the implicative-orthomodular widelattices are special cases of quantum-Wajsberg algebras., Comment: arXiv admin note: text overlap with arXiv:2401.04140

Starting from involutive BE algebras, we redefine the pre-MV and meta-MV algebras, by introducing the notion of pre-Wajsberg and meta-Wajsberg algebras, as generalizations of quantum-Wajsberg algebras. We characterize these algebras, we show that any pre-Wajsberg algebra is a meta-Wajsberg algebra, and we prove that an implicative-orthomodular algebra is a quantum-Wajsberg algebra if and only if it is a meta-Wajsberg algebra. It is shown that the commutative quantum-Wajsberg (implicative-orthomodular, pre-Wajsberg, meta-Wajsberg) algebras coincide with the Wajsberg algebras. We give conditions for the implicative-orthomodular, pre-Wajsberg and meta-Wajsberg algebras to be Wajsberg algebras. The commutative center of an implicative-orthomodular algebra is defined and studied, and it is proved that it is a Wajsberg subalgebra of the implicative-orthomodular algebra., Comment: arXiv admin note: text overlap with arXiv:2311.02483, arXiv:2310.13879, arXiv:2303.16481

We define the Wajsberg-center and the OML-center of a quantum-Wajsberg algebra, and study their structures. We prove that the Wajsberg-center is a Wajsberg subalgebra of a quantum-Wajsberg algebra, and that it is a distributive sublattice of its corresponding poset. If the quantum-Wajsberg algebra is quasi-linear, we show that the Wajsberg-center is a linearly ordered Wajsberg algebra. We also show that the lattice subreduct of the Wajsberg-center is a Kleene algebra. Furthermore, we prove that the OML-center is an orthomodular lattice, and the orthomodular lattices form a subvariety of the variety of quantum-Wajsberg algebras., Comment: arXiv admin note: text overlap with arXiv:2303.16481

Starting from involutive BE algebras, we redefine the orthomodular algebras, by introducing the notion of implicative-orthomodular algebras. We investigate properties of implicative-orthomodular algebras, and give characterizations of these algebras. Then we define and study the notions of filters and deductive systems, and characterize certain classes of filters. Furthermore, we introduce and characterize the commutative deductive systems in implicative-orthomodular algebras. We also show that any deductive system determines a congruence, and conversely, for any congruence we can define a deductive system in an implicative-orthomodular algebra. We define the quotient implicative-orthomodular algebra with respect to the congruence induced by a deductive system, and prove that a deductive system is commutative if and only if all deductive systems of the corresponding quotient algebra are commutative., Comment: arXiv admin note: text overlap with arXiv:2304.01845

We define and study the notions of q-deductive systems, p-deductive systems, deductive systems, maximal and strongly maximal q-deductive systems in quantum-Wajsberg algebras. We also introduce the notion of congruences induced by deductive systems of a quantum-Wajsberg algebra, and we show that there is a relationship between congruences and deductive systems. Furthermore, we define the quotient quantum-Wajsberg algebra with respect to a deductive system, and prove that the quotient quantum-Wajsberg algebra is locally finite if and only if the deductive system is strongly maximal. Finally, we define the weakly linear and quasi-linear quantum-Wajsberg algebras, and give a characterization of weakly linear quotient quantum-Wajsberg algebras.

We redefine the quantum-MV algebras starting from involutive BE algebras and we introduce and study the notion of quantum-Wajsberg algebras. We define the commutative quantum-Wajsberg algebras and we give a characterization of these structures. We also prove that any Wajsberg algebra is a quantum-Wajsberg algebra, and give conditions for quantum-Wajsberg algebras to be Wajsberg algebras. Furthermore, we prove that Wajsberg algebras are both quantum-Wajsberg algebras and quantum-B algebras. Finally, we provide certain conditions for quantum-MV algebras to be MV algebras.

The aim of this paper is to define and study the involutive and weakly involutive quantum B-algebras. We prove that any weakly involutive quantum B-algebra is a quantum B-algebra with pseudo-product. As an application, we introduce and investigate the notions of existential and universal quantifiers on involutive quantum B-algebras. It is proved that there is a one-to-one correspondence between the quantifiers on weakly involutive quantum B-algebras. One of the main results consists of proving that any pair of quantifiers is a monadic operator on weakly involutive quantum B-algebras. We investigate the relationship between quantifiers on bounded sup-commutative pseudo BCK-algebras and quantifiers on other related algebraic structures, such as pseudo MV-algebras and bounded Wajsberg hoops.

In this paper we define the monadic pseudo BE-algebras and investigate their properties. We prove that the existential and universal quantifiers of a monadic pseudo BE-algebra form a residuated pair. Special properties are studied for the particular case of monadic bounded commutative pseudo BE-algebras. Monadic classes of pseudo BE-algebras are investigated and it is proved that the quantifiers on bounded commutative pseudo BE-algebras are also quantifiers on the corresponding pseudo MV-algebras. The monadic deductive systems and monadic congruences of monadic pseudo BE-algebras are defined and their properties are studied. It is proved that, in the case of a monadic distributive commutative pseudo BE-algebra there is a one-to-one correspondence between monadic congruences and monadic deductive systems, and the monadic quotient pseudo BE-algebra algebra is also defined.

In this paper we define two types of implicative derivations on pseudo-BCI algebras, we investigate their properties and we give a characterization of regular implicative derivations of type II. We also define the notion of a $d$-invariant deductive system of a pseudo-BCI algebra $A$ proving that $d$ is a regular derivation of type II if and only if every deductive system on $A$ is $d$-invariant. It is proved that a pseudo-BCI algebra is $p$-semisimple if and only if the only regular derivation of type II is the identity map. Another main result consists of proving that the set of all implicative derivations of a $p$-semisimple pseudo-BCI algebra forms a commutative monoid with respect to function composition. Two types of symmetric derivations on pseudo-BCI algebras are also introduced and it is proved that in the case of $p$-semisimple pseudo-BCI algebras the sets of type II implicative derivations and type II symmetric derivations are equal.

In this paper we introduce the very true operators on pseudo-BCK algebras and we study their properties. We prove that the composition of two very true operators is a very true operator if and only if they commute. Moreover, given a very true bounded pseudo-BCK algebra $(A,v)$, we define the pseudo-BCK$_{vt,st}$ algebra by adding two truth-depressing hedges operators associated with $v$. We also define the very true deductive systems and the very true homomorphisms and we investigate their properties. Also, given a normal $v$-deductive system of a very true pseudo-BCK algebra $(A,v)$ we construct a very true operator on the quotient pseudo-BCK algebra $A/H$. We investigate the very true operators on some classes of pseudo-BCK algebras such as pseudo-BCK(pP) algebras, FL$_w$-algebras, pseudo-MTL algebras. Finally, we define the $Q$-Smarandache pseudo-BCK algebras and we introduce the notion of a very true operator on $Q$-Smarandache pseudo-BCK algebras.

In this this paper we introduce the notion of involutive filters of pseudo-hoops, and we emphasize their role in the probability theory on these structures. A characterization of involutive pseudo-hoops is given and their properties are investigated. We give characterizations of involutive filters of a bounded pseudo-hoop and we prove that in the case of bounded Wajsberg pseudo-hoops the notions of fantastic and involutive filters coincide. One of main results consists of proving that a normal filter $F$ of a bounded pseudo-hoop $A$ is involutive if and only if $A/F$ is an involutive pseudo-hoop. It is also proved that any Boolean filter of a bounded Wajsberg pseudo-hoop is involutive. The notions of state operators and state-morphism operators on pseudo-hoops are introduced and the relationship between these operators are investigated. For a bounded Wajsberg pseudo-hoop we prove that the kernel of any state operator is an involutive filter.

The aim of this paper is to introduce the notion of fantastic deductive systems on generalizations of fuzzy structures, and to emphasize their role in the probability theory on these algebras. We give a characterization of commutative pseudo-BE algebras and we generalize an axiom system consisting of four identities to the case of commutative pseudo-BE algebras. We define the fantastic deductive systems of pseudo-BE algebras and we investigate their properties. It is proved that, if a pseudo-BE(A) algebra $A$ is commutative, then all deductive systems of $A$ are fantastic. Moreover, we generalize the notions of measures, state-measures and measure-morphisms to the case of pseudo-BE algebras and we also prove that there is a one-to-one correspondence between the set of all Bosbach states on a bounded pseudo-BE algebra and the set of its state-measures. The notions of internal states and state-morphism operators on pseudo-BCK algebras are extended to the case of pseudo-BE algebras and we also prove that any type II state operator on a pseudo-BE algebra is a state-morphism operator on it. The notions of pseudo-valuation and commutative pseudo-valuation on pseudo-BE algebras are defined and investigated. For the case of commutative pseudo-BE algebras we prove that the two kind of pseudo-valuations coincide. Characterizations of pseudo-valuations and commutative pseudo-valuations are given. We show that the kernel of a Bosbach state (state-morphism, measure, type II state operator, pseudo-valuation) is a fantastic deductive system.

Pseudo equality algebras were initially introduced by Jenei and $\rm K\acute{o}r\acute{o}di$ as a possible algebraic semantic for fuzzy type theory, and they have been revised by Dvure\v{c}enskij and Zahiri under the name of JK-algebras. In this paper we define and study the commutative pseudo equality algebras. We give a characterization of commutative pseudo equality algebras and we prove that an invariant pseudo equality algebra is commutative if and only if its corresponding pseudo BCK(pC)-meet-semilattice is commutative. Other results consist of proving that every commutative pseudo equality algebra is a distributive lattice and every finite invariant commutative pseudo equality algebra is a symmetric pseudo equality algebra. We also introduce and investigate the commutative deductive systems of pseudo equality algebras. As applications of these notions and results we define and study the measures and measure-morphisms on pseudo equality algebras, we prove new properties of state pseudo equality algebras, and we introduce and investigate the pseudo-valuations on pseudo equality algebras., Comment: arXiv admin note: substantial text overlap with arXiv:1602.07828

In this paper we generalize the axiom systems given by M. Pa{\l}asi\'nski, B. Wo\'zniakowska and by W.H. Cornish for commutative BCK-algebras to the case of commutative pseudo BCK-algebras. A characterization of commutative pseudo BCK-algebras is also given. We define the commutative deductive systems of pseudo BCK-algebras and we generalize some results proved by Yisheng Huang for commutative ideals of BCI-algebras to the case of commutative deductive systems of pseudo BCK-algebras. We prove that a pseudo BCK-algebra $A$ is commutative if and only if all the deductive systems of $A$ are commutative. We show that a normal deductive system $H$ of a pseudo BCK-algebra $A$ is commutative if and only if $A/H$ is a commutative pseudo BCK-algebra. We introduce the notions of state operators and state-morphism operators on pseudo BCK-algebras, and we apply these results on commutative deductive systems to investigate the properties of these operators.

Pseudo equality algebras were initially introduced by Jenei and $\rm K\acute{o}r\acute{o}di$ as a possible algebraic semantic for fuzzy type theory, and they have been revised by Dvure\v censkij and Zahiri under the name of JK-algebras. The aim of this paper is to investigate the internal states and the state-morphisms on pseudo equality algebras. We define and study new classes of pseudo equality algebras, such as commutative, symmetric, pointed and compatible pseudo equality algebras. We prove that any internal state (state-morphism) on a pseudo equality algebra is also an internal state (state-morphism) on its corresponding pseudo BCK(pC)-meet-semilattice, and we prove the converse for the case of linearly ordered symmetric pseudo equality algebras. We also show that any internal state (state-morphism) on a pseudo BCK(pC)-meet-semilattice is also an internal state (state-morphism) on its corresponding pseudo equality algebra. The notion of a Bosbach state on a pointed pseudo equality algebra is introduced and proved that any Bosbach state on a pointed pseudo equality algebra is also a Bosbach state on its corresponding pointed pseudo BCK(pC)-meet-semilattice. For the case of an invariant pointed pseudo equality algebra, we show that the Bosbach states on the two structures coincide.

The concept of a state MV-algebra was firstly introduced by Flaminio and Montagna in \cite{FlMo0} and \cite{FlMo} as an MV-algebra with internal state as a unary operation. Di Nola and Dvure\v{c}enskij gave a stronger version of a state MV-algebra in \cite{DiDV1}, \cite{DiDV2}. In the present paper we introduce the notion of a state BL-algebra, or more precisely, a BL-algebra with internal state. We present different types of state BL-algebras, like strong state BL-algebras and state-morphism BL-algebras, and we study some classes of state BL-algebras. In addition, we give a sample of important examples of state BL-algebras and present some open problems.

In this paper, we extend the notions of states and measures presented in \cite{DvPu} to the case of pseudo-BCK algebras and study similar properties. We prove that, under some conditions, the notion of a state in the sense of \cite{DvPu} coincides with the Bosbach state, and we extend to the case of pseudo-BCK algebras some results proved by J. K\"uhr only for pseudo-BCK semilattices. We characterize extremal states, and show that the quotient pseudo-BCK algebra over the kernel of a measure can be embedded into the negative cone of an archimedean $\ell$-group. Additionally, we introduce a Borel state and using results by J. K\"uhr and D. Mundici from \cite{Kumu}, we prove a relationship between de Finetti maps, Bosbach states and Borel states.