Perfect residuated lattice ordered monoids

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Jiří Rachůnek ◽  
Dana Šalounová
Keyword(s):  
Probability Measure ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Heyting Algebras ◽  
Effect Algebras ◽  
Quantum Logics ◽  
Mv Algebras ◽  
Intuitionistic Logics

AbstractBounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).

scholarly journals Residuated Structures and Orthomodular Lattices

Studia Logica ◽  
2021 ◽  
Author(s):  
D. Fazio ◽  
A. Ledda ◽  
F. Paoli
Keyword(s):  
Algebraic Logic ◽  
Residuated Lattices ◽  
Quantum Structures ◽  
Heyting Algebras ◽  
Quantum Algebras ◽  
Orthomodular Lattices ◽  
Lattice Of Subvarieties ◽  
Residuated Structures ◽  
Common Framework ◽  
Mv Algebras

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

scholarly journals Injective and projective semimodules over involutive semirings

2021 ◽  
pp. 2250182
Author(s):  
Peter Jipsen ◽  
Sara Vannucci
Keyword(s):  
Crucial Role ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Term Equivalence ◽  
Involutive Residuated Lattice ◽  
Necessary And Sufficient ◽  
Mv Algebras

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval [Formula: see text] to be a subalgebra of an involutive residuated lattice, where [Formula: see text] is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.

Basic algebras, logics, trends and applications

2015 ◽  
Vol 08 (03) ◽  
pp. 1550040 ◽  
Author(s):  
Ivan Chajda
Keyword(s):  
Quantum Mechanics ◽  
20Th Century ◽  
19Th Century ◽  
Classical Logic ◽  
Boolean Algebras ◽  
Heyting Algebras ◽  
Fuzzy Logics ◽  
Orthomodular Lattices ◽  
Mv Algebras ◽  
Intuitionistic Logics

The classical logic was axiomatized algebraically by means of Boolean algebras in 19th century by George Boole. Similar attempts went on 20th century for algebraic axiomatization of non-classical logics, e.g. intuitionistic logics (Brouwer and Heyting algebras), many-valued logics (Łukasiewicz, Chang’s MV-algebras, Post algebras), the logic of quantum mechanics (orthomodular lattices and posets) and fuzzy logics (residuated lattices). In this paper, we are focused in a common generalization of MV-algebras and orthomodular lattices. The resulting algebras, called basic algebras, have surprisingly strong and interesting properties and they can be investigated in their own. The aim of the paper is to get an overview of results reached during the last decade.

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.

scholarly journals Nilpotent Elements of Residuated Lattices

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Shokoofeh Ghorbani ◽  
Lida Torkzadeh
Keyword(s):  
Residuated Lattice ◽  
Residuated Lattices ◽  
Nilpotent Elements ◽  
Mv Algebras

Some properties of the nilpotent elements of a residuated lattice are studied. The concept of cyclic residuated lattices is introduced, and some related results are obtained. The relation between finite cyclic residuated lattices and simple MV-algebras is obtained. Finally, the notion of nilpotent elements is used to define the radical of a residuated lattice.

scholarly journals Laterally complete and projective hulls of semilinear residuated lattices

10.29007/mmts ◽  
2018 ◽  
Author(s):  
José Gil-Férez ◽  
Antonio Ledda ◽  
Constantine Tsinakis
Keyword(s):  
Wide Class ◽  
Existence And Uniqueness ◽  
Residuated Lattice ◽  
Direct Limit ◽  
Residuated Lattices ◽  
Double Negation ◽  
Vector Lattices ◽  
Complete Vector ◽  
The Given ◽  
Mv Algebras

The existence of lateral completions of ℓ-groups is an old problem that was first solved, for conditionally complete vector lattices, by Nakano. The existence and uniqueness of lateral completions of representable ℓ-groups was first obtained as a consequence of the orthocompletions of Bernau, and later the proofs were simplified by Conrad, who also proved the existence and uniqueness of lateral completions of ℓ-groups with zero radical. Finally, Bernau solved the problem for ℓ-groups in general.In this work, we address the problem of the existence and uniqueness of lateral, projectable, and strongly projectable completions of residuated lattices. In particular, we push the methods of Conrad through to the case of the representable GMV-algebras.The leading idea is to construct, for any given semilinear residuated lattice, an orthocomplete extension such that the former is dense in the latter. This extension is obtained as the direct limit of a family of residuated lattices that are constructed using maximal partitions of the algebra of polars of the original residuated lattice.In order to complete the proof we still need another hypothesis, which is an abstraction of the condition of double negation in which commutativity and integrality have been dropped, and determines the wide class of Generalized MV-algebras. This, together with the density, allows us to obtain the completions of the given residuated lattice.

Splittings in varieties of logic

2021 ◽  
pp. 1-48
Author(s):  
Brian A. Davey ◽  
Tomasz Kowalski ◽  
Christopher J. Taylor
Keyword(s):  
Residuated Lattices ◽  
Heyting Algebras ◽  
General Lemma ◽  
Negative Results ◽  
Splitting Algebras

We study splittings or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the non-splitting lemma, which when combined with some variety-specific constructions, yields each of our negative results: the variety of commutative integral residuated lattices contains no splitting algebras, and in the varieties of double Heyting algebras, dually pseudocomplemented Heyting algebras and regular double [Formula: see text]-algebras the only splitting algebras are the two-element and three-element chains.

scholarly journals Essential extensions of filters in residuated lattices

2021 ◽  
Author(s):  
Masoud Haveshki
Keyword(s):  
Boolean Algebra ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Essential Extension ◽  
Mv Algebra ◽  
Essential Extensions

Abstract We define the essential extension of a filter in the residuated lattice A associated to an ideal of L(A) and investigate its related properties. We prove the residuated lattice A is a Boolean algebra, G(RL)-algebra or MV -algebra if and only if the essential extension of {1} associated to A \ P is a Boolean filter, G-filter or MV -filter (for all P ∈ SpecA), respectively. Also, some properties of lattice of essential extensions are studied.

scholarly journals Left residuated lattices induced by lattices with a unary operation

2019 ◽  
Vol 24 (2) ◽  
pp. 723-729
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Orthomodular Lattice ◽  
Residuated Lattice ◽  
Unary Operation ◽  
Residuated Lattices ◽  
Boolean Algebras ◽  
Natural Question ◽  
Similar Technique ◽  
Orthomodular Lattices

Abstract In a previous paper, the authors defined two binary term operations in orthomodular lattices such that an orthomodular lattice can be organized by means of them into a left residuated lattice. It is a natural question if these operations serve in this way also for more general lattices than the orthomodular ones. In our present paper, we involve two conditions formulated as simple identities in two variables under which this is really the case. Hence, we obtain a variety of lattices with a unary operation which contains exactly those lattices with a unary operation which can be converted into a left residuated lattice by use of the above mentioned operations. It turns out that every lattice in this variety is in fact a bounded one and the unary operation is a complementation. Finally, we use a similar technique by using simpler terms and identities motivated by Boolean algebras.

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