Recognizable sets of graphs: equivalent definitions and closure properties

1994 ◽  
Vol 4 (1) ◽  
pp. 1-32 ◽  
Author(s):  
Bruno Courcelle
Keyword(s):  
Algebraic Structure ◽  
Algebraic Structures ◽  
Closure Properties ◽  
Definition Of ◽  
Standard Consequence

The notion of a recognizable set of words, trees or graphs is relative to an algebraic structure on the set of words, trees or graphs respectively. We establish that several algebraic structures yield the same notion of a recognizable set of graphs. This notion is equivalent to that of a fully cutset-regular set of graphs introduced by Fellows and Abrahamson. We also establish that the class of recognizable sets of graphs is closed under the operations considered in these various equivalent definitions. This fact is not a standard consequence of the definition of recognizability.

Algebraic structures for pairwise comparison matrices: Consistency, social choices and Arrow’s theorem

2021 ◽  
Vol 71 (5) ◽  
pp. 1047-1062
Author(s):  
Giuseppina Barbieri ◽  
Antonio Boccuto ◽  
Gaetano Vitale
Keyword(s):  
Algebraic Structure ◽  
Pairwise Comparison ◽  
Consistency Index ◽  
General Definition ◽  
Fuzzy Approach ◽  
Algebraic Structures ◽  
Arrow’S Theorem ◽  
Arrow's Theorem ◽  
Definition Of

Abstract We present the algebraic structures behind the approaches used to work with pairwise comparison matrices and, in general, the representation of preferences. We obtain a general definition of consistency and a universal decomposition in the space of PCMs, which allow us to define a consistency index. Also Arrow’s theorem, which is presented in a general form, is relevant. All the presented results can be seen in the main formulations of PCMs, i.e., multiplicative, additive and fuzzy approach, by the fact that each of them is a particular interpretation of the more general algebraic structure needed to deal with these theories.

scholarly journals Neutrosophic Quotient Algebra

2019 ◽  
pp. 83-89
Author(s):  
Binu R ◽  
Keyword(s):  
Quotient Algebra ◽  
Neutrosophic Set ◽  
Algebraic Structures ◽  
Algebra Isomorphism ◽  
Algebraic Properties ◽  
Definition Of

The algebraic properties of neutrosphic ideals over algebra, isomorphism properties of neutrosophic ideal and neutrosophic modules over algebra are discussed in this paper. Some of the charactrisations of Neutrosophic quotient algebra are derived and the role of algebraic structures is studied in the context of neutrosophic set. This paper expands the definition of quotient algebra within the context of neutrosophical set.

On single valued neutrosophic sets and neutrosophic א-structures: Applications on algebraic structures (hyperstructures)

2020 ◽  
pp. 108-117
Author(s):  
Madeleine Al Al-Tahan ◽  
◽  
◽  
Bijan Davvaz
Keyword(s):  
Algebraic Structure ◽  
Algebraic Structures ◽  
Neutrosophic Sets

In this paper, we find a relationship between SVNS and neutrosophic N-structures and study it. Moreover, we apply our results to algebraic structures (hyperstructures) and prove that the results on neutrosophic N-substructure (subhyperstructure) of a given algebraic structure (hyperstructure) can be deduced from single valued neutrosophic algebraic structure (hyperstructure) and vice versa.

An approach to fuzzy multi-ideals of near rings

2021 ◽  
pp. 1-11
Author(s):  
Madeline Al Tahan ◽  
Sarka Hoskova-Mayerova ◽  
Bijan Davvaz
Keyword(s):  
Algebraic Structure ◽  
Algebraic Structures

In recent years, fuzzy multisets have become a subject of great interest for researchers and have been widely applied to algebraic structures including groups, rings, and many other algebraic structures. In this paper, we introduce the algebraic structure of fuzzy multisets as fuzzy multi-subnear rings (multi-ideals) of near rings. In this regard, we define different operations on fuzzy multi-ideals of near rings and we generalize some results known for fuzzy ideals of near rings to fuzzy multi-ideals of near rings.

scholarly journals Logic and Truth: Some Logics without Theorems

2008 ◽  
pp. 104-117
Author(s):  
Jayanta Sen ◽  
Mihir Kumar Chakraborty
Keyword(s):  
Algebraic Structure ◽  
Logical Consequence ◽  
The Other ◽  
Algebraic Structures ◽  
Sequent Calculi

Two types of logical consequence are compared: one, with respect to matrix and designated elements and the other with respect to ordering in a suitable algebraic structure. Particular emphasis is laid on algebraic structures in which there is no top-element relative to the ordering. The significance of this special condition is discussed. Sequent calculi for a number of such structures are developed. As a consequence it is re-established that the notion of truth as such, not to speak of tautologies, is inessential in order to define validity of an argument.

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 Translation, modulation and dilation systems in set-valued signal processing

2018 ◽  
Vol 10 (1) ◽  
pp. 143-164 ◽  
Author(s):  
H. Levent ◽  
Y. Yilmaz
Keyword(s):  
Signal Processing ◽  
Linear Space ◽  
Algebraic Structure ◽  
Compact Convex ◽  
Inner Product ◽  
Complex Numbers ◽  
Real Numbers ◽  
Aumann Integral ◽  
Definition Of ◽  
Interval Valued

In this paper, we investigate a very important function space consists of set-valued functions defined on the set of real numbers with values on the space of all compact-convex subsets of complex numbers for which the $p$th power of their norm is integrable. In general, this space is denoted by $L^{p}% (\mathbb{R},\Omega(\mathbb{C}))$ for $1\leq p<\infty$ and it has an algebraic structure named as a quasilinear space which is a generalization of a classical linear space. Further, we introduce an inner-product (set-valued inner product) on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ and we think it is especially important to manage interval-valued data and interval-based signal processing. This also can be used in imprecise expectations. The definition of inner-product on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is based on Aumann integral which is ready for use integration of set-valued functions and we show that the space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is a Hilbert quasilinear space. Finally, we give translation, modulation and dilation operators which are three fundational set-valued operators on Hilbert quasilinear space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$.

Algebraic Structure of Step Traces and Interval Traces

2020 ◽  
Vol 175 (1-4) ◽  
pp. 253-280
Author(s):  
Ryszard Janicki ◽  
Łukasz Mikulski
Keyword(s):  
Algebraic Structure ◽  
Formal Analysis ◽  
Concurrent Systems ◽  
Interval Order ◽  
Canonical Forms ◽  
Algebraic Structures ◽  
Main Subject

Traces and their extensions as comtraces, step traces and interval traces are quotient monoids over sequences or step sequences that play an important role in the formal analysis and verification of concurrent systems. Step traces are generalizations of comtraces and classical traces while interval traces are specialized traces that can deal with interval order semantics. The algebraic structures and their properties as projections, hidings, canonical forms and other invariants are very well established for traces and fairly well established for comtraces. For step traces and interval traces they are the main subject of this paper.

Some Subfields of Qp, and their Non-Standard Analogues

1974 ◽  
Vol 26 (02) ◽  
pp. 473-491
Author(s):  
Diana L. Dubrovsky
Keyword(s):  
Algebraic Structure ◽  
Recursion Theory ◽  
Algebraic Structures ◽  
Mathematical Structures ◽  
Closed Fields ◽  
Standard Models ◽  
Algebraically Closed Fields

The desire to study constructive properties of given mathematical structures goes back many years; we can perhaps mention L. Kronecker and B. L. van der Waerden, two pioneers in this field. With the development of recursion theory it was possible to make precise the notion of "effectively carrying out" the operations in a given algebraic structure. Thus, A. Frölich and J. C. Shepherdson [7] and M. O . Rabin [13] studied computable algebraic structures, i.e. structures whose operations can be viewed as recursive number theoretic relations. A. Robinson [18] and E. W. Madison [11] used the concepts of computable and arithmetically definable structures in order to establish the existence of what can be called non-standard analogues (in a sense that will be specified later) of certain subfields of R and C, the standard models for the theories of real closed and algebraically closed fields respectively.

Functionals defined by transfinite recursion

1965 ◽  
Vol 30 (2) ◽  
pp. 155-174 ◽  
Author(s):  
W. W. Tait
Keyword(s):  
Functional Equations ◽  
Partial Ordering ◽  
Closure Properties ◽  
Primitive Recursive Arithmetic ◽  
Primitive Recursion ◽  
Definition Of ◽  
Second Order Systems ◽  
Transfinite Recursion ◽  
Primitive Recursive ◽  
Elementary Closure

This paper deals mainly with quantifier-free second order systems (i.e., with free variables for numbers and functions, and constants for numbers, functions, and functionals) whose basic rules are those of primitive recursive arithmetic together with definition of functionals by primitive recursion and explicit definition. Precise descriptions are given in §2. The additional rules have the form of definition by transfinite recursion up to some ordinal ξ (where ξ is represented by a primitive recursive (p.r.) ordering). In §3 we discuss some elementary closure properties (under rules of inference and definition) of systems with recursion up to ξ. Let Rξ denote (temporarily) the system with recursion up to ξ. The main results of this paper are of two sorts:Sections 5–7 are concerned with less elementary closure properties of the systems Rξ. Namely, we show that certain classes of functional equations in Rη can be solved in Rη for some explicitly determined η < ε(η) (the least ε-number > ξ). The classes of functional equations considered all have roughly the form of definition by recursion on the partial ordering of unsecured sequences of a given functional F, or on some ordering which is obtained from this by simple ordinal operations. The key lemma (Theorem 1) needed for the reduction of these equations to transfinite recursion is simply a sharpening of the Brouwer-Kleene idea.

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