Strong NMV-algebras, commutative basic algebras and naBL-algebras

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Xiaohong Zhang
Keyword(s):  
Residuated Lattice ◽  
Basic Algebra ◽  
Commutative Basic Algebra ◽  
Ordered Groupoid ◽  
Commutative Residuated Lattice ◽  
The Relationship

AbstractThe aim of the paper is to investigate the relationship among NMV-algebras, commutative basic algebras and naBL-algebras (i.e., non-associative BL-algebras). First, we introduce the notion of strong NMV-algebra and prove that(1)a strong NMV-algebra is a residuated l-groupoid (i.e., a bounded integral commutative residuated lattice-ordered groupoid)(2)a residuated l-groupoid is commutative basic algebra if and only if it is a strong NMV-algebra.Secondly, we introduce the notion of NMV-filter and prove that a residuated l-groupoid is a strong NMV-algebra (commutative basic algebra) if and only if its every filter is an NMV-filter. Finally, we introduce the notion of weak naBL-algebra, and show that any strong NMV-algebra (commutative basic algebra) is weak naBL-algebra and give some counterexamples.

BCC-algebras and residuated partially-ordered groupoid

2013 ◽  
Vol 63 (3) ◽  
Author(s):  
Xiaohong Zhang
Keyword(s):  
Partially Ordered ◽  
Ordered Groupoid ◽  
The Relationship

AbstractThe aim of the paper is to investigate the relationship between BCC-algebras and residuated partially-ordered groupoids. We prove that an integral residuated partially-ordered groupoid is an integral residuated pomonoid if and only if it is a double BCC-algebra. Moreover, we introduce the notion of weakly integral residuated pomonoid, and give a characterization by the notion of pseudo-BCI algebra. Finally, we give a method to construct a weakly integral residuated pomonoid (pseudo-BCI algebra) from any bounded pseudo-BCK algebra with pseudo product and any group.

scholarly journals Adjoint Operations in Twist-Products of Lattices

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 253
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Residuated Lattice ◽  
Sufficient Conditions ◽  
Residuated Lattices ◽  
Double Negation ◽  
Antitone Involution ◽  
Binary Operations ◽  
Double Negation Law ◽  
Commutative Residuated Lattice

Given an integral commutative residuated lattices L=(L,∨,∧), its full twist-product (L2,⊔,⊓) can be endowed with two binary operations ⊙ and ⇒ introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a∈L we define a certain subset Pa(L) of L2. We characterize when Pa(L) is a sublattice of the full twist-product (L2,⊔,⊓). In this case Pa(L) together with some natural antitone involution ′ becomes a pseudo-Kleene lattice. If L is distributive then (Pa(L),⊔,⊓,′) becomes a Kleene lattice. We present sufficient conditions for Pa(L) being a subalgebra of (L2,⊔,⊓,⊙,⇒) and thus for ⊙ and ⇒ being a pair of adjoint operations on Pa(L). Finally, we introduce another pair ⊙ and ⇒ of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law, and we investigate when Pa(L) is closed under these new operations.

An example of a commutative basic algebra which is not an MV-algebra

2010 ◽  
Vol 60 (2) ◽  
Author(s):  
Michal Botur
Keyword(s):  
Subdirect Product ◽  
Lattice Structure ◽  
Basic Algebra ◽  
Commutative Basic Algebra ◽  
Mv Algebra ◽  
One To One ◽  
Open Question ◽  
Mv Algebras

AbstractMany algebras arising in logic have a lattice structure with intervals being equipped with antitone involutions. It has been proved in [CHK1] that these lattices are in a one-to-one correspondence with so-called basic algebras. In the recent papers [BOTUR, M.—HALAŠ, R.: Finite commutative basic algebras are MV-algebras, J. Mult.-Valued Logic Soft Comput. (To appear)]. and [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] we have proved that every finite commutative basic algebra is an MV-algebra, and that every complete commutative basic algebra is a subdirect product of chains. The paper solves in negative the open question posed in [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] whether every commutative basic algebra on the interval [0, 1] of the reals has to be an MV-algebra.

scholarly journals New topology in residuated lattices

2018 ◽  
Vol 16 (1) ◽  
pp. 1104-1127 ◽  
Author(s):  
L.C. Holdon
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Discrete Space ◽  
Regular Space ◽  
Quotient Topology ◽  
Uniform Topology ◽  
The Relationship ◽  
Mv Algebras

AbstractIn this paper, by using the notion of upsets in residuated lattices and defining the operator Da(X), for an upset X of a residuated lattice L we construct a new topology denoted by τa and (L, τa) becomes a topological space. We obtain some of the topological aspects of these structures such as connectivity and compactness. We study the properties of upsets in residuated lattices and we establish the relationship between them and filters. O. Zahiri and R. A. Borzooei studied upsets in the case of BL-algebras, their results become particular cases of our theory, many of them work in residuated lattices and for that we offer complete proofs. Moreover, we investigate some properties of the quotient topology on residuated lattices and some classes of semitopological residuated lattices. We give the relationship between two types of quotient topologies τa/F and $\begin{array}{} \displaystyle \mathop {{\tau _a}}\limits^ - \end{array}$. Finally, we study the uniform topology $\begin{array}{} \displaystyle {\tau _{\bar \Lambda }} \end{array}$ and we obtain some conditions under which $\begin{array}{} \displaystyle (L/J,{\tau _{\bar \Lambda }}) \end{array}$ is a Hausdorff space, a discrete space or a regular space ralative to the uniform topology. We discuss briefly the applications of our results on classes of residuated lattices such as divisible residuated lattices, MV-algebras and involutive residuated lattices and we find that any of this subclasses of residuated lattices with respect to these topologies form semitopological algebras.

Modal operators on bounded commutative residuated ℓ-monoids

2007 ◽  
Vol 57 (4) ◽  
Author(s):  
Jiří Rachůnek ◽  
Dana Šalounová
Keyword(s):  
Fuzzy Logic ◽  
Residuated Lattice ◽  
Heyting Algebras ◽  
Modal Operator ◽  
Modal Operators ◽  
Special Cases ◽  
Basic Fuzzy Logic ◽  
Commutative Residuated Lattice ◽  
Derived Algebras ◽  
Mv Algebras

AbstractBounded commutative residuated lattice ordered monoids (Rℓ-monoids) are a common generalization of, e.g., Heyting algebras and BL-algebras, i.e., algebras of intuitionistic logic and basic fuzzy logic, respectively. Modal operators (special cases of closure operators) on Heyting algebras were studied in [MacNAB, D. S.: Modal operators on Heyting algebras, Algebra Universalis 12 (1981), 5–29] and on MV-algebras in [HARLENDEROVÁ,M.—RACHŮNEK, J.: Modal operators on MV-algebras, Math. Bohem. 131 (2006), 39–48]. In the paper we generalize the notion of a modal operator for general bounded commutative Rℓ-monoids and investigate their properties also for certain derived algebras.

Separation axioms in \(L\)-fuzzifying supra-topology

2017 ◽  
Vol 8 (1) ◽  
pp. 67
Author(s):  
A. K. Mousa
Keyword(s):  
Residuated Lattice ◽  
Double Negation ◽  
Completely Distributive ◽  
Separation Axioms ◽  
Complete Residuated Lattice ◽  
Distributive Law ◽  
Double Negation Law ◽  
The Relationship

In this paper, we define and investigate the notions of \(L\)-separation axioms in \(L\)-fuzzifying supra-topology. Also, some of their characterizations and a systematic discussion on the relationship among these notions is gave in \(L\)-fuzzifying supra-topology where \(L\) is a complete residuated lattice. Sometimes we need more conditions on \(L\) such as the completely distributive law or that the "\(\wedge\)" is distributive over arbitrary joins or the double negation law as we illustrate through this paper. As applications of our work the corresponding results (see \cite{2, 13}) are generalized and new consequences are obtained.

Tips for Beginners: Graphing Inequalities with the y-Intercept

1991 ◽  
Vol 84 (4) ◽  
pp. 294-295
Author(s):  
Linda A. Morrell
Keyword(s):  
Basic Algebra ◽  
First Year ◽  
Mathematics Students ◽  
Algebra Students ◽  
The Right ◽  
Relationship Of ◽  
The Relationship

The confusion that often accompanies the topic of graphing inequalities came to my attention last spring as a result of my tutoring a student in basic algebra. Students are taught to shade to the left of the line, to the right of the line, and above or below the line. These techniques are valid methods for teaching graphing, but what is described as “to the right of the line” by one teacher may be referred to as “above the line” by another teacher. Strong first-year algebra students develop an intuitive approach to graphing and discern patterns in graphs and the relationship of the graph of the inequality to the graph of the equation having the same x- and y-intercepts. However, students who are not strong mathematics students may never see this relationship and develop only greater confusion!

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