Partial algebra + order-sorted algebra = galactic algebra

2005 ◽  
pp. 314-325 ◽  
Author(s):  
Aristide Mégrelis
Keyword(s):  
Partial Algebra

THE SEMILATTICE OF INNER EXTENSIONS OF A PARTIAL ALGEBRA

1987 ◽  
Vol 20 (1-2) ◽  
Author(s):  
W. Bartol ◽  
D. Niwinski ◽  
L. Rudak
Keyword(s):  
Partial Algebra

Implementing theorem provers in a purely functional style

1999 ◽  
Vol 9 (2) ◽  
pp. 147-166 ◽  
Author(s):  
KEITH HANNA
Keyword(s):  
Partial Algebra ◽  
Proof Assistant ◽  
Theorem Provers ◽  
Executable Specification ◽  
Efficient Representation

This paper discusses the principles of implementing an LCF-style proof assistant using a purely functional metalanguage. Two approaches are described; in both, signatures are treated as ordinary values, rather than as mutable components within an abstract datatype. The first approach treats the object logic as a partial algebra and represents it as a partial datatype, that is, a datatype in which the domains of the constructors are restricted by predicate functions. This results in a compact, executable specification of the logic. The second approach uses an abstract type to allow an efficient representation of the logic, whilst keeping the same interface. A case study describes how these principles were put into practice in implementing a fairly complex dependently-sorted logic.

On the semilattice of inner extensions of a fuzzy partial algebra

2002 ◽  
Vol 127 (3) ◽  
pp. 383-390 ◽  
Author(s):  
J. Casasnovas ◽  
M. Monserrat ◽  
F. Rosselló
Keyword(s):  
Partial Algebra

scholarly journals What Is the Sense in Logic and Philosophy of Language

2020 ◽  
Vol 49 (2) ◽  
Author(s):  
Urszula Wybraniec-Skardowska
Keyword(s):  
Philosophy Of Language ◽  
Partial Algebra ◽  
Algebraic Models ◽  
Natural Languages ◽  
Semantic Categories ◽  
Language L ◽  
Semantic Correspondence

In the paper, various notions of the logical semiotic sense of linguistic expressions – namely, syntactic and semantic, intensional and extensional – are considered and formalised on the basis of a formal-logical conception of any language L characterised categorially in the spirit of certain Husserl's ideas of pure grammar, Leśniewski-Ajdukiewicz's theory of syntactic/semantic categories and, in accordance with Frege's ontological canons, Bocheński's and some of Suszko's ideas of language adequacy of expressions of L. The adequacy ensures their unambiguous syntactic and semantic senses and mutual, syntactic and semantic correspondence guaranteed by the acceptance of a postulate of categorial compatibility of syntactic and semantic (extensional and intensional) categories of expressions of L. This postulate defines the unification of these three logical senses. There are three principles of compositionality which follow from this postulate: one syntactic and two semantic ones already known to Frege. They are treated as conditions of homomorphism of partial algebra of L into algebraic models of L: syntactic, intensional and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, obviously, an idealisation. The syntactic and semantic unambiguity of its expressions is not, of course, a feature of natural languages, but every syntactically and semantically ambiguous expression of such languages may be treated as a schema representing all of its interpretations that are unambiguous expressions.

scholarly journals Unsolid and fluid strong varieties of partial algebras

2006 ◽  
Vol 2006 ◽  
pp. 1-19
Author(s):  
S. Busaman ◽  
K. Denecke
Keyword(s):  
Partial Algebra ◽  
The Other ◽  
Turing Machines ◽  
Left Hand ◽  
Recursive Functions ◽  
Right Hand ◽  
Algebraic Description ◽  
Partial Recursive Functions ◽  
Partial Algebras ◽  
The Right

A partial algebra𝒜=(A;(fiA)i∈I)consists of a setAand an indexed set(fiA)i∈Iof partial operationsfiA:Ani⊸→A. Partial operations occur in the algebraic description of partial recursive functions and Turing machines. A pair of termsp≈qover the partial algebra𝒜is said to be a strong identity in𝒜if the right-hand side is defined whenever the left-hand side is defined and vice versa, and both are equal. A strong identityp≈qis called a strong hyperidentity if when the operation symbols occurring inpandqare replaced by terms of the same arity, the identity which arises is satisfied as a strong identity. If every strong identity in a strong variety of partial algebras is satisfied as a strong hyperidentity, the strong variety is called solid. In this paper, we consider the other extreme, the case when the set of all strong identities of a strong variety of partial algebras is invariant only under the identical replacement of operation symbols by terms. This leads to the concepts of unsolid and fluid varieties and some generalizations.

scholarly journals NeutroAlgebra is a Generalization of Partial Algebra

2020 ◽  
pp. 08-17
Author(s):  
Florentin Smarandache ◽  
Keyword(s):  
Partial Algebra ◽  
The Other ◽  
Partial Operation ◽  
Whole Space ◽  
Partial Algebras ◽  
Concept Attribute

In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019 research referred to NeutroAlgebras and AntiAlgebras (also called NeutroAlgebraic Structures and respectively AntiAlgebraic Structures). Let be an item (concept, attribute, idea, proposition, theory, etc.). Through the process of neutrosphication, we split the nonempty space we work on into three regions {two opposite ones corresponding to and , and one corresponding to neutral (indeterminate) (also denoted ) between the opposites}, which may or may not be disjoint – depending on the application, but they are exhaustive (their union equals the whole space). A NeutroAlgebra is an algebra which has at least one NeutroOperation or one NeutroAxiom (axiom that is true for some elements, indeterminate for other elements, and false for the other elements). A Partial Algebra is an algebra that has at least one Partial Operation, and all its Axioms are classical (i.e. axioms true for all elements). Through a theorem we prove that NeutroAlgebra is a generalization of Partial Algebra, and we give examples of NeutroAlgebras that are not Partial Algebras. We also introduce the NeutroFunction (and NeutroOperation).

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