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Categorical Pullbacks

Marco Riccardi (2015)

Formalized Mathematics

The main purpose of this article is to introduce the categorical concept of pullback in Mizar. In the first part of this article we redefine homsets, monomorphisms, epimorpshisms and isomorphisms [7] within a free-object category [1] and it is shown there that ordinal numbers can be considered as categories. Then the pullback is introduced in terms of its universal property and the Pullback Lemma is formalized [15]. In the last part of the article we formalize the pullback of functors [14] and it...

Cauchy-like functional equation based on a class of uninorms

Feng Qin (2015)

Kybernetika

Commuting is an important property in any two-step information merging procedure where the results should not depend on the order in which the single steps are performed. In the case of bisymmetric aggregation operators with the neutral elements, Saminger, Mesiar and Dubois, already reduced characterization of commuting n -ary operators to resolving the unary distributive functional equations. And then the full characterizations of these equations are obtained under the assumption that the unary...

Cayley's Theorem

Artur Korniłowicz (2011)

Formalized Mathematics

The article formalizes the Cayley's theorem saying that every group G is isomorphic to a subgroup of the symmetric group on G.

Characteristic of Rings. Prime Fields

Christoph Schwarzweller, Artur Korniłowicz (2015)

Formalized Mathematics

The notion of the characteristic of rings and its basic properties are formalized [14], [39], [20]. Classification of prime fields in terms of isomorphisms with appropriate fields (ℚ or ℤ/p) are presented. To facilitate reasonings within the field of rational numbers, values of numerators and denominators of basic operations over rationals are computed.

Choice functions and well-orderings over the infinite binary tree

Arnaud Carayol, Christof Löding, Damian Niwinski, Igor Walukiewicz (2010)

Open Mathematics

We give a new proof showing that it is not possible to define in monadic second-order logic (MSO) a choice function on the infinite binary tree. This result was first obtained by Gurevich and Shelah using set theoretical arguments. Our proof is much simpler and only uses basic tools from automata theory. We show how the result can be used to prove the inherent ambiguity of languages of infinite trees. In a second part we strengthen the result of the non-existence of an MSO-definable well-founded...

Circumcenter, Circumcircle and Centroid of a Triangle

Roland Coghetto (2016)

Formalized Mathematics

We introduce, using the Mizar system [1], some basic concepts of Euclidean geometry: the half length and the midpoint of a segment, the perpendicular bisector of a segment, the medians (the cevians that join the vertices of a triangle to the midpoints of the opposite sides) of a triangle. We prove the existence and uniqueness of the circumcenter of a triangle (the intersection of the three perpendicular bisectors of the sides of the triangle). The extended law of sines and the formula of the radius...

Classes of fuzzy filters of residuated lattice ordered monoids

Jiří Rachůnek, Dana Šalounová (2010)

Mathematica Bohemica

The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (monoids) are common generalizations of BL -algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding...

Communication with www in Czech

Lukáš Svoboda, Luboš Popelínský (2004)

Kybernetika

This paper describes UIO, a multi–domain question–answering system for the Czech language that looks for answers on the web. UIO exploits two fields, namely natural language interface to databases and question answering. In its current version, UIO can be used for asking questions about train and coach timetables, cinema and theatre performances, about currency exchange rates, name–days and on the Diderot Encyclopaedia. Much effort have been made into making addition of a new domain very easy. No...

Compactness and Löwenheim-Skolem properties in categories of pre-institutions

Antonino Salibra, Giuseppe Scollo (1993)

Banach Center Publications

The abstract model-theoretic concepts of compactness and Löwenheim-Skolem properties are investigated in the "softer" framework of pre-institutions [18]. Two compactness results are presented in this paper: a more informative reformulation of the compactness theorem for pre-institution transformations, and a theorem on natural equivalences with an abstract form of the first-order pre-institution. These results rely on notions of compact transformation, which are introduced as arrow-oriented generalizations...

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