Partial Menger algebras of terms

The superposition operation [Formula: see text] [Formula: see text] [Formula: see text], maps to each [Formula: see text]-tuple of [Formula: see text]-ary operations on a set [Formula: see text] an [Formula: see text]-ary operation on [Formula: see text] and satisfies the so-called superassociative law, a generalization of the associative law. The corresponding algebraic structures are Menger algebras of rank [Formula: see text]. A partial algebra of type [Formula: see text] which satisfies the superassociative law as weak identity is said to be a partial Menger algebra of rank [Formula: see text]. As a generalization of linear terms we define [Formula: see text]-terms as terms where each variable occurs at most [Formula: see text]-times. It will be proved that [Formula: see text]-ary [Formula: see text]-terms form partial Menger algebras of rank [Formula: see text]. In this paper, some algebraic properties of partial Menger algebras such as generating systems, homomorphic images and freeness are investigated. As generalization of hypersubstitutions and linear hypersubstitutions we consider [Formula: see text]-hypersubstitutions.

What Is the Sense in Logic and Philosophy of Language

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.

NeutroAlgebra is a Generalization of Partial Algebra

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).

Unsolid and fluid strong varieties of partial algebras

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.