De Morgan Algebras Research Articles

A Neutrosophic Recommender System for Medical Diagnosis Based on Algebraic Neutrosophic Measures

Neutrosophic set has the ability to handle uncertain, incomplete, inconsistent, indeterminate information in a more accurate way. In this paper, we proposed a neutrosophic recommender system to predict the diseases based on neutrosophic set which includes single-criterion neutrosophic recommender system (SC-NRS) and multi-criterion neutrosophic recommender system (MC-NRS). Further, we investigated some algebraic operations of neutrosophic recommender system such as union, complement, intersection, probabilistic sum, bold sum, bold intersection, bounded difference, symmetric difference, convex linear sum of min and max operators, Cartesian product, associativity, commutativity and distributive. Based on these operations, we studied the algebraic structures such as lattices, Kleen algebra, de Morgan algebra, Brouwerian algebra, BCK algebra, Stone algebra and MV algebra. In addition, we introduced several types of similarity measures based on these algebraic operations and studied some of their theoretic properties. Moreover, we accomplished a prediction formula using the proposed algebraic similarity measure. We also proposed a new algorithm for medical diagnosis based on neutrosophic recommender system. Finally to check the validity of the proposed methodology, we made experiments on the datasets Heart, RHC, Breast cancer, Diabetes and DMD. At the end, we presented the MSE and computational time by comparing the proposed algorithm with the relevant ones such as ICSM, DSM, CARE, CFMD, as well as other variants namely Variant 67, Variant 69, and Varian 71 both in tabular and graphical form to analyze the efficiency and accuracy. Finally we analyzed the strength of all 8 algorithms by ANOVA statistical tool.

The Lattices of Kernel Ideals in Pseudocomplemented De Morgan Algebras

In a pseudocomplemented de Morgan algebra, it is shown that the set of kernel ideals is a complete Heyting lattice, and a necessary and sufficient condition that the set of kernel ideals is boolean (resp. Stone) is derived. In particular, a characterization of a de Morgan Heyting algebra whose congruence lattice is boolean (resp. Stone) is given.

Semisimple varieties of implication zroupoids

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In fact, in 1934, Bernstein (Trans Am Math Soc 36:876---884, 1934) gave a system of axioms for Boolean algebras in terms of implication only. Though his original axioms were not equational, a quick look at his axioms would reveal that if one adds a constant, then it is not hard to translate his system of axioms into an equational one. Recently, in 2012, the second author of this paper extended this modified Bernstein's theorem to De Morgan algebras (see Sankappanavar, Sci Math Jpn 75(1):21---50, 2012). Indeed, it is shown in Sankappanavar (Sci Math Jpn 75(1):21---50, 2012) that the varieties of De Morgan algebras, Kleene algebras, and Boolean algebras are term-equivalent, respectively, to the varieties, $$\mathbf {DM}$$DM, $$\mathbf {KL}$$KL, and $$\mathbf {BA}$$BA whose defining axioms use only the implication $$\rightarrow $$ź and the constant 0. The fact that the identity, herein called (I), occurs as one of the two axioms in the definition of each of the varieties $$\mathbf {DM}$$DM, $$\mathbf {KL}$$KL and $$\mathbf {BA}$$BA motivated the second author of this paper to introduce, and investigate, the variety $$\mathbf {I}$$I of implication zroupoids, generalizing De Morgan algebras. These investigations are continued by the authors of the present paper in Cornejo and Sankappanavar (Implication zroupoids I, 2015), wherein several new subvarieties of $$\mathbf {I}$$I are introduced and their relationships with each other and with the varieties studied in Sankappanavar (Sci Math Jpn 75(1):21---50, 2012) are explored. The present paper is a continuation of Sankappanavar (Sci Math Jpn 75(1):21---50, 2012) and Cornejo and Sankappanavar (Implication zroupoids I, 2015). The main purpose of this paper is to determine the simple algebras in $$\mathbf {I}$$I. It is shown that there are exactly five (nontrivial) simple algebras in $$\mathbf {I}$$I. From this description we deduce that the semisimple subvarieties of $$\mathbf {I}$$I are precisely the subvarieties of the variety generated by these simple I-zroupoids and that they are locally finite. It also follows that the lattice of semisimple subvarieties of $$\mathbf {I}$$I is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain.

Generalized Fuzzy σ - Algebra and Generalized Fuzzy Measure on Soft Sets

In this paper, we define soft fuzzy De Morgan algebra, soft generalized fuzzy σ algebra and soft generalized fuzzy measure. Related to these definitions, we give some examples, lemmas and interesting results on soft generalized fuzzy measure space.

Order in Implication Zroupoids

The variety $${\mathbf{I}}$$I of implication zroupoids (using a binary operation $${\to}$$� and a constant 0) was defined and investigated by Sankappanavar (Scientia Mathematica Japonica 75(1):21---50, 2012), as a generalization of De Morgan algebras. Also, in Sankappanavar (Scientia Mathematica Japonica 75(1):21---50, 2012), several subvarieties of $${\mathbf{I}}$$I were introduced, including the subvariety $${\mathbf{I_{2,0}}}$$I2,0, defined by the identity: $${x^{\prime \prime}\approx x}$$x��x, which plays a crucial role in this paper. Some more new subvarieties of $${\mathbf{I}}$$I are studied in Cornejo and Sankappanavar (Algebra Univ, 2015) that includes the subvariety $${\mathbf{SL}}$$SL of semilattices with a least element 0. An explicit description of semisimple subvarieties of $${\mathbf{I}}$$I is given in Cornejo and Sankappanavar (Soft Computing, 2015). It is a well known fact that there is a partial order (denote it by $${\sqsubseteq}$$�) induced by the operation �, both in the variety $${\mathbf{SL}}$$SL of semilattices with a least element and in the variety $${\mathbf{DM}}$$DM of De Morgan algebras. As both $${\mathbf{SL}}$$SL and $${\mathbf{DM}}$$DM are subvarieties of $${\mathbf{I}}$$I and the definition of partial order can be expressed in terms of the implication and the constant, it is but natural to ask whether the relation $${\sqsubseteq}$$� on $${\mathbf{I}}$$I is actually a partial order in some (larger) subvariety of $${\mathbf{I}}$$I that includes both $${\mathbf{SL}}$$SL and $${\mathbf{DM}}$$DM. The purpose of the present paper is two-fold: Firstly, a complete answer is given to the above mentioned problem. Indeed, our first main theorem shows that the variety $${\mathbf{I_{2,0}}}$$I2,0 is a maximal subvariety of $${\mathbf{I}}$$I with respect to the property that the relation $${\sqsubseteq}$$� is a partial order on its members. In view of this result, one is then naturally led to consider the problem of determining the number of non-isomorphic algebras in $${\mathbf{I_{2,0}}}$$I2,0 that can be defined on an n-element chain (herein called $${\mathbf{I_{2,0}}}$$I2,0-chains), n being a natural number. Secondly, we answer this problem in our second main theorem which says that, for each $${n \in \mathbb{N}}$$n�N, there are exactly n nonisomorphic $${\mathbf{I_{2,0}}}$$I2,0-chains of size n.

Model checking computation tree logic over finite lattices

Multi-valued model-checking is an extension of classical model-checking used to verify the properties of systems with uncertain information content. It is used in systems where the semantic domain of both the models of the systems and the specification is a De Morgan algebra or more abstract structure such as semirings. A common feature of De Morgan algebras and semirings is that they satisfy the distributive law, which is crucial for all of the multi-valued model-checking algorithms developed so far. In this paper, we study computation tree logic with membership values in a finite lattice, which are called multi-valued computation tree logic (MCTL). We introduce three semantics for MCTL over the multi-valued Kripke structure: path semantics, fixpoint semantics, and algebraic semantics, and prove that they coincide if and only if the underlying lattice is distributive. Furthermore, we provide model-checking algorithms with respect to the three semantics. Our algorithms show that MCTL model-checking problems with respect to the fixpoint and algebraic semantics can be solved in polynomial time in the size of the state space of the multi-valued Kripke structure, the size of the lattice, and the length of the formula, while MCTL model-checking problem with respect to the path semantics is in PSPACE. We also provide a lower bound for the MCTL model-checking problem with respect to the path semantics.

Two Axiomatizations of Nelson Algebras

Nelson algebras were first studied by Rasiowa and Białynicki- Birula [1] under the name N-lattices or quasi-pseudo-Boolean algebras. Later, in investigations by Monteiro and Brignole [3, 4], and [2] the name “Nelson algebras” was adopted - which is now commonly used to show the correspondence with Nelson’s paper [14] on constructive logic with strong negation. By a Nelson algebra we mean an abstract algebra 〈L, T, -, ¬, →, ⇒, ⊔, ⊓〉 where L is the carrier, − is a quasi-complementation (Rasiowa used the sign ~, but in Mizar “−” should be used to follow the approach described in [12] and [10]), ¬ is a weak pseudo-complementation → is weak relative pseudocomplementation and ⇒ is implicative operation. ⊔ and ⊓ are ordinary lattice binary operations of supremum and infimum. In this article we give the definition and basic properties of these algebras according to [16] and [15]. We start with preliminary section on quasi-Boolean algebras (i.e. de Morgan bounded lattices). Later we give the axioms in the form of Mizar adjectives with names corresponding with those in [15]. As our main result we give two axiomatizations (non-equational and equational) and the full formal proof of their equivalence. The second set of equations is rather long but it shows the logical essence of Nelson lattices. This formalization aims at the construction of algebraic model of rough sets [9] in our future submissions. Section 4 contains all items from Th. 1.2 and 1.3 (and the itemization is given in the text). In the fifth section we provide full formal proof of Th. 2.1 p. 75 [16].

Admissibility via natural dualities

It is shown that admissible clauses and quasi-identities of quasivarieties generated by a single finite algebra, or equivalently, the quasiequational and universal theories of their free algebras on countably infinitely many generators, may be characterized using natural dualities. In particular, axiomatizations are obtained for the admissible clauses and quasi-identities of bounded distributive lattices, Stone algebras, Kleene algebras and lattices, and De Morgan algebras and lattices.

Algebraic and topological structures based on novel soft set relations

The algebraic and topological structures based on a new kind of soft set relations, Z-soft set relations, are characterized. Here we introduce the Z-soft set relations and provide further investigations to them. Several new operations of Z-soft set relations are employed to investigate the fundamental properties of Z-soft set relations. In particular, it is shown that the collection of all Z-soft set relations over a soft set gives rise to commutative idempotent hemirings, MV-algebras, Boolean algebras, De Morgan algebras, Kleene algebras and Stone algebras with respect to the new operations defined in this paper. Further, Z-soft set relation topologies are studied by employing Z-anti-reflexive interior operators and Z-reflexive closure operators of Z-soft set relations. It is noteworthy that intrinsic connections between Z-anti-reflexive interior operators and Z-reflexive closure operators have been established.

δ-Ideals in MS-Algebras

The concepts of δ-Ideals and principal δ-Ideals are introduced in an MS-algebra and many properties of these ideals are studied. It is observed that the class of all δ-Ideals forms a complete distributive lattice and the class of all principal δ-Ideals forms a de Morgan algebra. A characterization of δ-Ideals in terms of principal δ-Ideals is given. Finally, many properties of δ-Ideals are studied with respect to homomorphisms.

Discrete Duality for Tense Łukasiewicz–Moisil Algebras

In 2007, tense Łukasiewicz–Moisil algebras were introduced by Diaconescu and Georgescu as an algebraic counterpart of tense n–valued Moisil logic. These algebras constitute a generalization of tense algebras. In this paper we describe a discrete duality for tense Łukasiewicz–Moisil algebras bearing in mind the results indicated by Dzik, Orlowska and van Alten in 2006, for De Morgan algebras.

Congruence Lattices of Symmetric Extended De Morgan Algebras

In this paper, we characterize the congruence lattice of a symmetric extended De Morgan algebra $$L$$ . We show that the congruence lattice of the algebra $$L$$ is a pseudocomplemented lattice, and that such a congruence lattice is a Stone lattice if and only if the lattice of the compact congruences on $$L$$ forms a complete Boolean lattice. In particular, we prove that the congruence lattice of $$L$$ is a Boolean lattice if and only if, it is a relative Stone lattice, which is the case, if and only if $$L$$ is finite.

Congruence properties of pseudocomplemented De Morgan algebras

In this paper we first describe the Priestley duality for pseudocomplemented De Morgan algebras by combining the known dualities of distributive p-algebras due to Priestley and for De Morgan algebras due to Cornish and Fowler. We then use it to characterize congruence-permutability, principal join property, and the property of having only principal congruences for pseudocomplemented De Morgan algebras. The congruence-uniform pseudocomplemented De Morgan algebras are also described.

De Morgan Functions and Free De Morgan Algebras

Abstract It is commonly known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the bounded lattice of monotone Boolean functions of n variables. In this paper, we introduce the concept of De Morgan function and prove that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. This is a solution of the problem suggested by B. I. Plotkin.

Boole–De Morgan Algebras and Quasi-De Morgan Functions

In this paper we establish a Stone-type and a Birkhoff-type representation theorems for Boole–De Morgan algebras and prove that the free Boole–De Morgan algebra on n free generators is isomorphic to the Boole–De Morgan algebra of quasi-De Morgan functions of n variables. Also we introduce the concept of Zhegalkin polynomials for quasi-De Morgan functions and consider the representation problem of those functions by polynomials.

Dualities for modal N4-lattices

We introduce a new Priestley-style topological duality for N4-lattices, which are the algebraic counterpart of paraconsistent Nelson logic. Our duality differs from the existing one, due to S. Odintsov, in that we only rely on Esakia duality for Heyting algebras and not on the duality for De Morgan algebras of Cornish and Fowler. A major advantage of our approach is that we obtain a simple description for our topological structures, which allows us to extend the duality to other algebraic structures such as N4-lattices with monotonic modal operators, and also to provide a neighborhood semantics for the non-normal modal logic corresponding to these algebras.

Matrix Characterization of 4-Ary Algebraic Operations of Idempotent Algebras

Let (U; F) be an idempotent algebra. There is an r-ary essentially algebraic operation in F where there is not any (r + 3)-ary algebraic operation depending on at least r + 1 variables. In this paper, we prove that the set of all 4-ary algebraic operations of this algebras forms a finite De Morgan algebra, and then we characterize this De Morgan algebra.

Super-De Morgan functions and free De Morgan quasilattices

AbstractA De Morgan quasilattice is an algebra satisfying hyperidentities of the variety of De Morgan algebras (lattices). In this paper we give a functional representation of the free n-generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on nfree generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of nvariables.

De Morgan Algebras with a Quasi-Stone Operator

We investigate the class of those algebras (L; o, *) in which (L; o) is a de Morgan algebra, (L; *) is a quasi-Stone algebra, and the operations $${x \mapsto x^{\circ}}$$ x ? x ? and $${x \mapsto x^{*}}$$ x ? x ? are linked by the identity x**o = x*o*. We show that such an algebra is subdirectly irreducible if and only if its congruence lattice is either a 2-element chain or a 3-element chain. In particular, there are precisely eight non-isomorphic subdirectly irreducible Stone de Morgan algebras.

Subdirectly irreducible algebras with hyperidentities of the variety of De Morgan algebras

The paper characterizes the class of subdirectly irreducible algebras satisfying hyperidentities of the variety of De Morgan algebras. Such algebras are called subdirectly irreducible De Morgan quasilattices. The suggested characterization is quite close to that of the classical case of subdirectly irreducible DeMorgan algebras.