scholarly journals Layered Predicates

1992 ◽  
Vol 21 (423) ◽  
Author(s):  
Flemming Nielson ◽  
Hanne Riis Nielson
Keyword(s):  
Equivalence Relations ◽  
Substitution Property ◽  
Structural Induction ◽  
Major Application ◽  
Partial Equivalence Relations ◽  
Definition Of

We review the concept of logical relations and how they interact with structural induction, furthermore we give examples of their use, and of particular interest is the combination with the PER-idea (partial equivalence relations). This is then generalized to Kripke logical relations; the major application is to show that in combination with the PER-idea this solves the problem of establishing a substitution property in a manner oonducive to structural induction. Finally we introduce the concept of Kripke-layered predicates; this allows a modular definition of predicates and supports a methodology of ''proofs in stages'' where each stage forcuses on only one aspect and thus is more manageable. All of these techniques have been tested and refined in ''realistic applications'' that have been documented elsewhere.

scholarly journals The Meaning of Types From Intrinsic to Extrinsic Semantics

2000 ◽  
Vol 7 (32) ◽  
Author(s):  
John C. Reynolds
Keyword(s):  
Lambda Calculus ◽  
Denotational Semantics ◽  
Equivalence Relations ◽  
Typed Lambda Calculus ◽  
Partial Equivalence Relations ◽  
Definition Of

A definition of a typed language is said to be "intrinsic" if it assigns<br />meanings to typings rather than arbitrary phrases, so that ill-typed<br />phrases are meaningless. In contrast, a definition is said to be "extrinsic"<br />if all phrases have meanings that are independent of their typings,<br />while typings represent properties of these meanings.<br />For a simply typed lambda calculus, extended with recursion, subtypes,<br />and named products, we give an intrinsic denotational semantics<br />and a denotational semantics of the underlying untyped language. We<br />then establish a logical relations theorem between these two semantics,<br />and show that the logical relations can be "bracketed" by retractions<br />between the domains of the two semantics. From these results, we<br />derive an extrinsic semantics that uses partial equivalence relations.

Fundamental relations and identities of fuzzy hyperalgebras

2021 ◽  
pp. 1-10
Author(s):  
Narjes Firouzkouhi ◽  
Abbas Amini ◽  
Chun Cheng ◽  
Mehdi Soleymani ◽  
Bijan Davvaz
Keyword(s):  
Universal Algebra ◽  
Equivalence Relation ◽  
Polynomial Function ◽  
Equivalence Relations ◽  
Fundamental Relation ◽  
Term Function ◽  
Biological Phenomena ◽  
Definition Of

Inspired by fuzzy hyperalgebras and fuzzy polynomial function (term function), some homomorphism properties of fundamental relation on fuzzy hyperalgebras are conveyed. The obtained relations of fuzzy hyperalgebra are utilized for certain applications, i.e., biological phenomena and genetics along with some elucidatory examples presenting various aspects of fuzzy hyperalgebras. Then, by considering the definition of identities (weak and strong) as a class of fuzzy polynomial function, the smallest equivalence relation (fundamental relation) is obtained which is an important tool for fuzzy hyperalgebraic systems. Through the characterization of these equivalence relations of a fuzzy hyperalgebra, we assign the smallest equivalence relation α i 1 i 2 ∗ on a fuzzy hyperalgebra via identities where the factor hyperalgebra is a universal algebra. We extend and improve the identities on fuzzy hyperalgebras and characterize the smallest equivalence relation α J ∗ on the set of strong identities.

scholarly journals Neurocosmetics in Skincare—The Fascinating World of Skin–Brain Connection: A Review to Explore Ingredients, Commercial Products for Skin Aging, and Cosmetic Regulation

Cosmetics ◽  
2021 ◽  
Vol 8 (3) ◽  
pp. 66
Author(s):  
Vito Rizzi ◽  
Jennifer Gubitosa ◽  
Paola Fini ◽  
Pinalysa Cosma
Keyword(s):  
Nervous System ◽  
Inflammatory Responses ◽  
Skin Aging ◽  
Cosmetic Products ◽  
Sensitive Skin ◽  
Functional Ingredients ◽  
Major Application ◽  
Regulatory Aspect ◽  
Brain Connection ◽  
Definition Of

The “modern” cosmetology industry is focusing on research devoted to discovering novel neurocosmetic functional ingredients that could improve the interactions between the skin and the nervous system. Many cosmetic companies have started to formulate neurocosmetic products that exhibit their activity on the cutaneous nervous system by affecting the skin’s neuromediators through different mechanisms of action. This review aims to clarify the definition of neurocosmetics, and to describe the features of some functional ingredients and products available on the market, with a look at the regulatory aspect. The attention is devoted to neurocosmetic ingredients for combating skin stress, explaining the stress pathways, which are also correlated with skin aging. “Neuro-relaxing” anti-aging ingredients derived from plant extracts and neurocosmetic strategies to combat inflammatory responses related to skin stress are presented. Afterwards, the molecular basis of sensitive skin and the suitable neurocosmetic ingredients to improve this problem are discussed. With the aim of presenting the major application of Botox-like ingredients as the first neurocosmetics on the market, skin aging is also introduced, and its theory is presented. To confirm the efficacy of the cosmetic products on the market, the concept of cosmetic claims is discussed.

scholarly journals L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1169-1188 ◽  
Author(s):  
ROMAN SAUER
Keyword(s):  
Probability Space ◽  
Betti Numbers ◽  
Free Action ◽  
Countable Group ◽  
Equivalence Relations ◽  
Dimension Theory ◽  
Discrete Groups ◽  
Type Ii ◽  
Orbit Equivalent ◽  
Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

scholarly journals Lattice Theory of Generalized Partitions

1959 ◽  
Vol 11 ◽  
pp. 97-106 ◽  
Author(s):  
Juris Hartmanis
Keyword(s):  
Equivalence Relation ◽  
Lattice Theory ◽  
Equivalence Relations ◽  
Unified Theory ◽  
Fixed Set ◽  
Higher Types ◽  
Definition Of ◽  
Partition Lattices

In (1) the lattice of all equivalence relations on a set S was studied and many important properties were established. In (2) and (3) the lattice of all geometries on a set S was studied and it was shown to be a universal lattice which shares many properties with the lattice of equivalence relations on S. In this paper we shall give the definition of a partition of type n and investigate the lattice formed by all partitions of type n on a fixed set S. It will be seen that a partition of type one on S can be considered as an equivalence relation on S and similarly a partition of type two on S can be considered as a geometry on S as defined in (2). Thus we shall obtain a unified theory of lattices of equivalence relations, lattices of geometries and partition lattices of higher types.

scholarly journals Cladistic hypotheses as degree of equivalence relational structures: implications for three-item statements

2021 ◽  
Author(s):  
Valentin Rineau ◽  
Stéphane Prin
Keyword(s):  
Phylogenetic Trees ◽  
Phylogenetic Reconstruction ◽  
Equivalence Relations ◽  
Strong Connection ◽  
Relational Structures ◽  
Degree Of Equivalence ◽  
Axiomatic Definition ◽  
Finite Set ◽  
Definition Of ◽  
Supertree Methods

AbstractThree-item statements, as minimal informative rooted binary phylogenetic trees on three items, are the minimal units of cladistic information. Their importance for phylogenetic reconstruction, consensus and supertree methods relies on both (i) the fact that any cladistic tree can always be decomposed into a set of three-item statements, and (ii) the possibility, at least under some conditions, to build a new cladistic tree by combining all or part of the three-item statements deduced from several prior cladistic trees. In order to formalise such procedures, several k-adic rules of inference, i.e., rules that allow us to deduce at least one new three-item statement from exactly k other ones, have been identified. However, no axiomatic background has been proposed, and it remains unknown if a particular k-adic rule of inference can be reduced to more basic rules. In order to solve this problem, we propose here to define three-item statements in terms of degree of equivalence relations. Given both the axiomatic definition of the latter and their strong connection to hierarchical classifications, we establish a list of the most basic properties for three-item statements. With such an approach, we show that it is possible to combine five three-item statements from basic rules although they are not combinable only from dyadic rules. Such a result suggests that all higher k-adic rules are well reducible to a finite set of simpler rules.

Least fixpoints of endofunctors of cartesian closed categories

1993 ◽  
Vol 3 (2) ◽  
pp. 229-257 ◽  
Author(s):  
J. Lambek
Keyword(s):  
Propositional Calculus ◽  
Second Order ◽  
Equivalence Relations ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Congruence Relations ◽  
Partial Recursive Functions ◽  
Cartesian Closed Categories ◽  
Partial Equivalence Relations ◽  
Cartesian Closed

Least fixpoints are constructed for finite coproducts of definable endofunctors of Cartesian closed categories that have weak polynomial products and joint equalizers of arbitrary families of pairs of parallel arrows. Both conditions hold in PER, the category whose objects are partial equivalence relations on N, and whose arrows are partial recursive functions. Weak polynomial products exist in any cartesian closed category with a finite number of objects as well as in any model of second order polymorphic lambda calculus: that is, in the proof theory of any second order positive intuitionistic propositional calculus, but such a category need not have equalizers. However, any finite coproduct of definable endofunctors of a cartesian closed category with weak polynomial products will have a least fixpoint in a larger category with equalizers whose objects are right ideals (or sieves) of modulo certain congruence relations, and whose arrows are induced from .

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