Commutative deductive systems of pseudo BCK-algebras

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

Commutative pseudo equality algebras

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:1602.0782

State pseudo equality algebras

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

Some properties of pseudo-BCK- and pseudo-BCI-algebras

Pseudo-BCI-algebras generalize both BCI-algebras and pseudo-BCK-algebras, which are a non-commutative generalization of BCK-algebras. In this paper, following [J.G. Raftery and C.J. van Alten, Residuation in commutative ordered monoids with minimal zero, Rep. Math. Log. 34 (2000) 23-57], we show that pseudo-BCI-algebras are the residuation subreducts of semi-integral residuated po-monoids and characterize those pseudo-BCI-algebras which are direct products of pseudo-BCK-algebras and groups (regarded as pseudo-BCI-algebras). We also show that the quasivariety of pseudo-BCI-algebras is relatively congruence modular; in fact, we prove that this holds true for all relatively point regular quasivarieties which are relatively ideal determined, in the sense that the kernels of relative congruences can be described by means of ideal terms

Involutive filters of pseudo-hoops

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

Pseudo BCI-algebras with derivations

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

On commutative weak BCK-algebras

The class of weak BCK-algebras is obtained by weakening one of standard BCK axioms. It is known that every weak BCK-algebra is completely determined by the structure of its initial segments. We review several natural classes of commutative weak BCK-algebras, prove that they are equationally definable, and show that the order duals of a multitude of algebras with implication known in the literature in connection with various quantum logics are, in fact, commutative weak BCK-algebras belonging to that or other of these classes. We also characterize initial segments of algebras in each of the classes as lattices equipped with a suitable kind of complementation. In particular, commutative weak BCK-algebras are just those meet semilattices with the least element in which all initial segments are non-distributive de Morgan lattices.Comment: LaTeX2e; 18 pages, no figures. Text revised and slightly extended (numeration of theorems in Sect.6 is changed), misprints corrected, Theorem 6.2 improved, list of references updated. This paper will not be published in full; [29] is its shortened versio

Further remarks on an order for quantum observables

S. Gudder and, later, S. Pulmanova and E. Vincekova, have studied in two recent papers a certain ordering of bounded self-adjoint operators on a Hilbert space. We present some further results on this ordering and show that some structure theorems of the ordered set of operators can be obtained in a more abstract setting of posets having the upper bound property and equipped with a certain orthogonality relation.Comment: LaTeX2e; 13 pages, no figures. Submitted to Mathematica Slovaca. Corrected misprints (in particular, in Example 3 on p.12), proof of Theorem 4(b) simplified, an argument improved at the end of Sect.4, Theorem 12 extended, its proof is now more transparen

Monadic pseudo BE-algebras

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

Measures, states and de Finetti maps on pseudo-BCK algebras

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