scholarly journals Equational bases for subvarieties of double MS-algebras

1989 ◽  
Vol 31 (1) ◽  
pp. 1-16 ◽  
Author(s):  
T. S. Blyth ◽  
A. S. A. Noor ◽  
J. C. Varlet
Keyword(s):  
Distributive Lattice ◽  
Universal Algebra ◽  
Lattice Theory ◽  
Unary Operation ◽  
Basic Properties ◽  
Ockham Algebras

An MS-algebra is an algebra (L; ∨, ∧, ∘, 0, 1) of type (2, 2, 1, 0, 0) such that (L; ∨, ∧, 0, 1) is a distributive lattice with smallest element 0 and greatest element 1, and x ↦ x∘ is a unary operation such that l∘ = 0, x ≤ x∘∘ for all x ∈ L, and (x ∧ y)∘ = x∘ ∨ y∘ for all x, y ∈ L. These algebras belong to the class of Ockham algebras introduced by Berman [3]; see also [2,10,15]. A double MS-algebra is an algebra (L, ∨, ∧, ∘, +, 0, 1) of type (2, 2, 1, 1, 0, 0) such that (L, ∘) and (Ld, +) are MS-algebras, where Ld denotes the dual of L, and the operations ∘, + are linked by the identities x∘+ = x∘∘ and x+∘ = x++. We refer to [5, 6, 7, 8] for the basic properties of MS-algebras and double MS-algebras. Concerning the latter, the properties x∘∘∘ = x∘, x+++ = x+, and x∘ ≤ x+will be used frequently. The class of double MS-algebras is congruencedistributive and consequently the results of [13] can be applied. As to general results in lattice theory and universal algebra, the reader may consult [1, 9, 12].

scholarly journals The ideal lattice of an MS-algebra

1988 ◽  
Vol 30 (2) ◽  
pp. 137-143 ◽  
Author(s):  
T. S. Blyth ◽  
J. C. Varlet
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Unary Operation ◽  
Stone Algebra ◽  
Ideal Lattice ◽  
Prime Filter ◽  
De Morgan Algebra ◽  
De Morgan Algebras ◽  
The Ideal

Recently we introduced the notion of an MS-algebra as a common abstraction of a de Morgan algebra and a Stone algebra [2]. Precisely, an MS-algebra is an algebra 〈L; ∧, ∨ ∘, 0, 1〉 of type 〈2, 2, 1, 0, 0〉 such that 〈L; ∧, ∨, 0, 1〉 is a distributive lattice with least element 0 and greatest element 1, and x → x∘ is a unary operation such that x ≤ x∘∘, (x ∧ y)∘ = x∘ ∨ y∘ and 1∘ = 0. It follows that ∘ is a dual endomorphism of L and that L∘∘ = {x∘∘ x ∊ L} is a subalgebra of L that is called the skeleton of L and that belongs to M, the class of de Morgan algebras. Clearly, theclass MS of MS-algebras is equational. All the subvarieties of MS were described in [3]. The lattice Λ (MS) of subvarieties of MS has 20 elements (see Fig. 1) and its non-trivial part (we exclude T, the class of one-element algebras) splits into the prime filter generated by M, that is [M, M1], the prime ideal generated by S, that is [B, S], and the interval [K, K2 ∨ K3].

On EMV-Semirings

2019 ◽  
Vol 69 (4) ◽  
pp. 739-752 ◽  
Author(s):  
R. A. Borzooei ◽  
M. Shenavaei ◽  
A. Di Nola ◽  
O. Zahiri
Keyword(s):  
Distributive Lattice ◽  
Algebraic Structure ◽  
Maximal Ideal ◽  
Boolean Algebras ◽  
Algebraic Extension ◽  
Theoretic Approach ◽  
Basic Properties ◽  
Mv Algebra ◽  
Definition Of ◽  
Generalized Boolean Algebras

Abstract The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that any EMV-semiring is a distributive lattice. Moreover, we define an EMV-semiring homomorphism and show that the categories of EMV-semirings and the category of EMV-algebras are isomorphic. We also define the concepts of GI-simple and DLO-semiring and prove that every EMV-semiring is a GI-simple and a DLO-semiring. Finally, we propose a representation for EMV-semirings, which proves that any EMV-semiring is either an MV-semiring or can be embedded into an MV-semiring as a maximal ideal.

scholarly journals Congruences on a Distributive Lattice

1954 ◽  
Vol 10 (2) ◽  
pp. 76-77
Author(s):  
H. A. Thueston
Keyword(s):  
Distributive Lattice ◽  
Lattice Theory ◽  
Distributive Lattices ◽  
The Subject ◽  
The Many ◽  
Finite Distributive Lattices

Among the many papers on the subject of lattices I have not seen any simple discussion of the congruences on a distributive lattice. It is the purpose of this note to give such a discussion for lattices with a certain finiteness. Any distributive lattice is isomorphic with a ring of sets (G. Birkhoff, Lattice Theory, revised edition, 1948, p. 140, corollary to Theorem 6); I take the case where the sets are finite. All finite distributive lattices are covered by this case.

Frontal operators in distributive lattices with a generalized implication

2015 ◽  
Vol 08 (03) ◽  
pp. 1550039
Author(s):  
Sergio A. Celani ◽  
Hernán J. San Martín
Keyword(s):  
Distributive Lattice ◽  
Unary Operation ◽  
Duke Math ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Bounded Distributive Lattice ◽  
Modal Lattices ◽  
Weak Heyting Algebras ◽  
San Martín

We introduce a family of extensions of bounded distributive lattices. These extensions are obtained by adding two operations: an internal unary operation, and a function (called generalized implication) that maps pair of elements to ideals of the lattice. A bounded distributive lattice with a generalized implication is called gi-lattice in [J. E. Castro and S. A. Celani, Quasi-modal lattices, Order 21 (2004) 107–129]. The main goal of this paper is to introduce and study the category of frontal gi-lattices (and some subcategories of it). This category can be seen as a generalization of the category of frontal weak Heyting algebras (see [S. A. Celani and H. J. San Martín, Frontal operators in weak Heyting algebras, Studia Logica 100(1–2) (2012) 91–114]). In particular, we study the case of frontal gi-lattices where the generalized implication is defined as the annihilator (see [B. A. Davey, Some annihilator conditions on distributive lattices, Algebra Universalis 4(1) (1974) 316–322; M. Mandelker, Relative annihilators in lattices, Duke Math. J. 37 (1970) 377–386]). We give a Priestley’s style duality for each one of the new classes of structures considered.

Recognizable languages in divisibility monoids

2001 ◽  
Vol 11 (6) ◽  
pp. 743-770 ◽  
Author(s):  
MANFRED DROSTE ◽  
DIETRICH KUSKE
Keyword(s):  
Distributive Lattice ◽  
Lattice Theory ◽  
Free Monoid ◽  
Rank Function ◽  
Commutative Monoids ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem

We define the class of divisibility monoids that arise as quotients of the free monoid Σ* modulo certain equations of the form ab = cd. These form a much larger class than free partially commutative monoids, and we show, under certain assumptions, that the recognizable languages in these divisibility monoids coincide with c-rational languages. The proofs rely on Ramsey's theorem, distributive lattice theory and on Hashigushi's rank function generalized to these monoids. We obtain Ochmański's theorem on recognizable languages in free partially commutative monoids as a consequence.

Free Involutorial Completely Simple Semigroups

1985 ◽  
Vol 37 (2) ◽  
pp. 271-295 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Maximal Subgroup ◽  
Universal Algebra ◽  
Simple Semigroup ◽  
Binary Operation ◽  
Unary Operation ◽  
Completely Simple Semigroup ◽  
Completely Simple Semigroups ◽  
Image Position ◽  
Completely Simple

An involution x → x* of a semigroup S is an antiautomorphism of S of order at most 2, that is (xy)* = y*x* and x** = x for all x, y ∊ S. In such a case, S is called an involutorial semigroup if regarded as a universal algebra with the binary operation of multiplication and the unary operation *. If S is also a completely simple semigroup, regarded as an algebra with multiplication and the unary operation x → x−1 of inversion (x−1 is the inverse of x in the maximal subgroup of S containing x), then (S, −1, *), or simply S, is an involutorial completely simple semigroup. All such S form a variety determined by the identities above concerning * andwhere x0 = xx−1.

scholarly journals Lattices in Social Networks with Influence

2015 ◽  
Vol 17 (01) ◽  
pp. 1540004
Author(s):  
Michel Grabisch ◽  
Agnieszka Rusinowska
Keyword(s):  
Social Networks ◽  
Social Network ◽  
Distributive Lattice ◽  
Dynamic Model ◽  
Influence Function ◽  
Lattice Theory ◽  
General Analysis ◽  
Aggregation Functions ◽  
Action Model ◽  
Boolean Lattices

We present an application of lattice theory to the framework of influence in social networks. The contribution of the paper is not to derive new results, but to synthesize our existing results on lattices and influence. We consider a two-action model of influence in a social network in which agents have to make their yes–no decision on a certain issue. Every agent is preliminarily inclined to say either "yes" or "no", but due to influence by others, the agent's decision may be different from his original inclination. We discuss the relation between two central concepts of this model: Influence function and follower function. The structure of the set of all influence functions that lead to a given follower function appears to be a distributive lattice. We also consider a dynamic model of influence based on aggregation functions and present a general analysis of convergence in the model. Possible terminal classes to which the process of influence may converge are terminal states (the consensus states and nontrivial states), cyclic terminal classes and unions of Boolean lattices.

scholarly journals Stone Lattices

2015 ◽  
Vol 23 (4) ◽  
pp. 387-396 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Lattice Theory ◽  
Cartesian Product ◽  
Unary Operation ◽  
Binary Operations ◽  
Pseudocomplemented Lattice ◽  
Stone Lattice ◽  
Current Article ◽  
Boolean Lattices ◽  
Pseudocomplemented Lattices ◽  
Natural Way

Summary The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense. The core of the paper is of course the idea of Stone identity $$a^* \sqcup a^{**} = {\rm{T}},$$ which is fundamental for us: Stone lattices are those lattices L, which are distributive, bounded, and satisfy Stone identity for all elements a ∈ L. Stone algebras were introduced by Grätzer and Schmidt in [18]. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined a * as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices. All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of p 2 for arbitrary prime number p (Section 6). At the end we formalize the notion of the Stone lattice B [2] (of pairs of elements a, b of B such that a ⩽ b) constructed as a sublattice of B 2, where B is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices. Our formalization was inspired by [17], and is an important step in formalizing Jouni Järvinen Lattice theory for rough sets [19], so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423–426. The description of handling complemented structures in Mizar [6] can be found in [12]. The current article together with [15] establishes the formal background for algebraic structures which are important for [10], [16] by means of mechanisms of merging theories as described in [11].

Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space

2020 ◽  
Vol 39 (3) ◽  
pp. 4515-4531 ◽  
Author(s):  
Mohammed Atef ◽  
Ahmed Mostafa Khalil ◽  
Sheng-Gang Li ◽  
A.A. Azzam ◽  
Abd El Fattah El Atik
Keyword(s):  
Rough Set ◽  
Basic Properties ◽  
Fundamental Properties ◽  
Approximation Operators ◽  
Neighborhood Rough Set ◽  
Type 3 ◽  
The Relationship

In this paper, we generalize three types of rough set models based on j-neighborhood space (i.e, type 1 j-neighborhood rough set, type 2 j-neighborhood rough set, and type 3 j-neighborhood rough set), and investigate some of their basic properties. Also, we present another three types of rough set models based on j-adhesion neighborhood space (i.e, type 4 j-adhesion neighborhood rough set, type 5 j-adhesion neighborhood rough set, and type 6 j-adhesion neighborhood rough set). The fundamental properties of approximation operators based on j-adhesion neighborhood space are established. The relationship between the properties of these types is explained. Finally, according to j-adhesion neighborhood space, we give a comparison between the Yao’s approach and our approach.

scholarly journals Many-one reductions and the category of multivalued functions

2015 ◽  
Vol 27 (3) ◽  
pp. 376-404
Author(s):  
ARNO PAULY
Keyword(s):  
Distributive Lattice ◽  
Complexity Theory ◽  
Systematic Investigation ◽  
Complexity Classes ◽  
Computable Analysis ◽  
Open Questions ◽  
Basic Theme ◽  
Carry Over ◽  
Multivalued Functions

Multivalued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the monikersearch problemsleading to complexity classes such as PPAD and PLS being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees).A more general understanding is possible, if the category-theoretic properties of multivalued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multivalued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g. Cook, Karp, Weihrauch).Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multivalued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multivalued functions.

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