Algebraic equivalences over fuzzy quantities
Algebraic Numbers
This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to the polynomial...
Algebraic properties of pre-logics
Algebras on subintervals of BL-algebras, pseudo BL-algebras and bounded residuated -monoids
Algèbre du calcul propositionnel trivalent de Heyting
Algèbres de Lukasiewicz symétriques
Algorithmische Transformation von Papierstrukturen - Ein Beitrag zur Verallgemeinerung des Konstruktionsbegriffes
All General Solutions of Finite Equations
All Liouville Numbers are Transcendental
In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real...
All reproductive solutions of finite equations.
Alternative mathematical model of the natural language semantics using first-order fuzzy logic [Abstract of thesis]
Altitude, Orthocenter of a Triangle and Triangulation
We introduce the altitudes of a triangle (the cevians perpendicular to the opposite sides). Using the generalized Ceva’s Theorem, we prove the existence and uniqueness of the orthocenter of a triangle [7]. Finally, we formalize in Mizar [1] some formulas [2] to calculate distance using triangulation.
Ambiguity and stratification
Amenability, extreme amenability, model-theoretic stability, and dependence property in integral logic
This paper has three parts. First, we study and characterize amenable and extremely amenable topological semigroups in terms of invariant measures using integral logic. We prove definability of some properties of a topological semigroup such as amenability and the fixed point on compacta property. Second, we define types and develop local stability in the framework of integral logic. For a stable formula ϕ, we prove definability of all complete ϕ-types over models and deduce from this the fundamental...
An admissibility criterion for inference rules with metavariables in the modal logic S4.
An algebraic and Kripke-style approach to a certain extension of intuitionistic logic [Book]
An algebraic approach to the Heyting-Brouwer predicate calculus
An algebraic completeness proof for Kleene's 3-valued logic
We introduce Kleene's 3-valued logic in a language containing, besides the Boolean connectives, a constant for the undefined truth value, so in developing semantics we can switch from the usual treatment based on DM-algebras to the narrower class of DMF-algebras (De Morgan algebras with a single fixed point for negation). A sequent calculus for Kleene's logic is introduced and proved complete with respect to threevalent semantics. The completeness proof is based on a version of the prime ideal...
An algorithm of calculation of lower solutions of fuzzy relational equations.
This paper deals with the problem of the determination of lower solutions of fuzzy relational equations. An algorithm of calculation of such a solution is presented.
An application of logical-probabilistic expressions to the realization of stochastic automata