FREE MONOID THEORY: MAXIMALITY AND COMPLETENESS IN ARBITRARY SUBMONOIDS

2003 ◽  
Vol 13 (05) ◽  
pp. 507-516 ◽  
Author(s):  
JEAN NÉRAUD ◽  
CARLA SELMI
Keyword(s):  
Free Monoid ◽  
Famous Result ◽  
Weak Completeness ◽  
Topological Density

In this paper, we discuss the different notions of local topological density for subsets of the free monoid A*. We introduce the notion of weak completeness for a set X, relatively to an arbitrary submonoid M of A*. For the so-called strongly M-thin codes, we establish that weak completeness is equivalent to maximality in M. This constitutes a new generalization of a famous result due to Schützenberger.

Internal categoricity and truth

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Lower Order ◽  
Higher Order ◽  
Mathematical Truth ◽  
Object Language ◽  
Famous Result ◽  
Famous Paper ◽  
Truth Value ◽  
Truth Operator

This chapter considers whether internal categoricity can be used to leverage any claims about mathematical truth. We begin by noting that internal categoricity allows us to introduce a truth-operator which gives an object-language expression to the supervaluationist semantics. In this way, the univocity discussed in previous chapters might seem to secure an object-language expression of determinacy of truth-value; but this hope falls short, because such truth-operators must be carefully distinguished from truth-predicates. To introduce these truth-predicates, we outline an internalist attitude towards model theory itself. We then use this to illuminate the cryptic conclusions of Putnam's justly-famous paper ‘Models and Reality’. We close this chapter by presenting Tarski’s famous result that truth for lower-order languages can be defined in higher-order languages.

FINITELY BASED WORDS

2000 ◽  
Vol 10 (04) ◽  
pp. 457-480 ◽  
Author(s):  
OLGA SAPIR
Keyword(s):  
Sufficient Conditions ◽  
Basis Property ◽  
Free Monoid ◽  
Finite Basis ◽  
Finitely Based ◽  
Finite Basis Property ◽  
Letter Alphabet ◽  
Finite Language ◽  
Rees Quotient ◽  
The Ideal

Let W be a finite language and let Wc be the closure of W under taking subwords. Let S(W) denote the Rees quotient of a free monoid over the ideal consisting of all words that are not in Wc. We call W finitely based if the monoid S(W) is finitely based. Although these semigroups have easy structure they behave "generically" with respect to the finite basis property [6]. In this paper, we describe all finitely based words in a two-letter alphabet. We also find some necessary and some sufficient conditions for a set of words to be finitely based.

A weak completeness theorem for infinite valued first-order logic

1963 ◽  
Vol 28 (1) ◽  
pp. 43-50 ◽  
Author(s):  
L. P. Belluce ◽  
C. C. Chang
Keyword(s):  
Completeness Theorem ◽  
Unit Interval ◽  
Order Logic ◽  
First Order Logic ◽  
Order System ◽  
First Order ◽  
Recursively Enumerable ◽  
Weak Completeness ◽  
Truth Value

This paper contains some results concerning the completeness of a first-order system of infinite valued logicThere are under consideration two distinct notions of completeness corresponding to the two notions of validity (see Definition 3) and strong validity (see Definition 4). Both notions of validity, whether based on the unit interval [0, 1] or based on linearly ordered MV-algebras, use the element 1 as the designated truth value. Originally, it was thought by many investigators in the field that one should be able to prove that the set of valid sentences is recursively enumerable. It was first proved by Rutledge in [9] that the set of valid sentences in the monadic first-order infinite valued logic is recursively enumerable.

scholarly journals Disjunctive languages on a free monoid

1977 ◽  
Vol 34 (2) ◽  
pp. 123-129 ◽  
Author(s):  
H.J. Shyr
Keyword(s):  
Free Monoid

On the strong semantical completeness of the intuitionistic predicate calculus

1968 ◽  
Vol 33 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Richmond H. Thomason
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
First Order ◽  
Semantic Tableaux ◽  
Weak Completeness ◽  
Skolem Theorem ◽  
Completeness Theorems

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

Γ-LANGUAGES AND Γ-AUTOMATA

2021 ◽  
Vol 10 (9) ◽  
pp. 3185-3194
Author(s):  
Anjeza Krakulli
Keyword(s):  
Free Monoid ◽  
Standard Theory

The aim of this paper is to extend the notion of an automaton as a triple made of a set of states, a free monoid on some set, and an action of this monoid on the set of states, to the case where the free monoid is replaced by a free Γ-monoid, and the action is replaced by the action of this Γ-monoid on the set of states. We call the respective triple a Γ-automaton. This concept leads to another new concept, that of a Γ-language, which is a subset of a free Γ-monoid. Also, we define recognizable Γ-languages and prove that they are exactly those Γ-languages that are recognized by a finite Γ-automaton. In the end, in analogy with the standard theory, we relate the recognizability of a Γ-language with the concept of division of semigroups.

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