A Representation Theorem for Geometric Morphisms

2009 ◽  
Vol 18 (6) ◽  
pp. 573-583 ◽  
Author(s):  
Christopher F. Townsend
Keyword(s):  
Representation Theorem ◽  
Geometric Morphisms

Flat functors and classifying toposes

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
General Theory ◽  
Geometric Theory ◽  
Small Category ◽  
Independent Interest ◽  
Yoneda Lemma ◽  
Geometric Morphisms ◽  
The Way

This chapter develops a general theory of extensions of flat functors along geometric morphisms of toposes; the attention is focused in particular on geometric morphisms between presheaf toposes induced by embeddings of categories and on geometric morphisms to the classifying topos of a geometric theory induced by a small category of set-based models of the latter. A number of general results of independent interest are established on the way, including developments on colimits of internal diagrams in toposes and a way of representing flat functors by using a suitable internalized version of the Yoneda lemma. These general results will be instrumental for establishing in Chapter 6 the main theorem characterizing the class of geometric theories classified by a presheaf topos and for applying it.

scholarly journals Study of Two Families of Generalized Yager’s Implications for Describing the Structure of Generalized (h,e)-Implications

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1490
Author(s):  
Raquel Fernandez-Peralta ◽  
Sebastia Massanet ◽  
Arnau Mir
Keyword(s):  
Representation Theorem ◽  
Internal Factors ◽  
Fuzzy Implication ◽  
The Family

In this study, we analyze the family of generalized (h,e)-implications. We determine when this family fulfills some of the main additional properties of fuzzy implication functions and we obtain a representation theorem that describes the structure of a generalized (h,e)-implication in terms of two families of fuzzy implication functions. These two families can be interpreted as particular cases of the (f,g) and (g,f)-implications, which are two families of fuzzy implication functions that generalize the well-known f and g-generated implications proposed by Yager through a generalization of the internal factors x and 1x, respectively. The behavior and additional properties of these two families are also studied in detail.

scholarly journals A representation theorem for the linear quasi-differential equation(py′)′+qy=0

2000 ◽  
Vol 23 (8) ◽  
pp. 579-584
Author(s):  
J. G. O'Hara
Keyword(s):  
Differential Equation ◽  
Comparison Theorem ◽  
Simple Proof ◽  
Representation Theorem ◽  
Second Order ◽  
Order Linear

We establish a representation forqin the second-order linear quasi-differential equation(py′)′+qy=0. We give a number of applications, including a simple proof of Sturm's comparison theorem.

A completeness theorem for higher order logics

2000 ◽  
Vol 65 (2) ◽  
pp. 857-884 ◽  
Author(s):  
Gábor Sági
Keyword(s):  
Set Theory ◽  
Representation Theorem ◽  
Completeness Theorem ◽  
Higher Order ◽  
Algebraic Solution ◽  
Algebraic Proof ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Strong Representation ◽  
Algebraic Counterpart

AbstractHere we investigate the classes of representable directed cylindric algebras of dimension α introduced by Németi [12]. can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, “purely cylindric algebraic” proof for the following theorems of Németi: (i) is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one can obtain a strong representation theorem for if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories.

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