A new proof of the congruence lattice representation theorem

1976 ◽  
Vol 6 (1) ◽  
pp. 269-275 ◽  
Author(s):  
Pavel Pudlák
Keyword(s):  
Congruence Lattice ◽  
Representation Theorem ◽  
Lattice Representation

A MINIMAL CONGRUENCE LATTICE REPRESENTATION FOR

2020 ◽  
Vol 108 (3) ◽  
pp. 332-340
Author(s):  
ROGER BUNN ◽  
DAVID GROW ◽  
MATT INSALL ◽  
PHILIP THIEM
Keyword(s):  
Congruence Lattice ◽  
Dihedral Group ◽  
Unary Algebra ◽  
Left Translation ◽  
Lattice Representation

Let $p$ be an odd prime. The unary algebra consisting of the dihedral group of order $2p$, acting on itself by left translation, is a minimal congruence lattice representation of $\mathbb{M}_{p+1}$.

scholarly journals Lattice initial segments of the hyperdegrees

2010 ◽  
Vol 75 (1) ◽  
pp. 103-130 ◽  
Author(s):  
Richard A. Shore ◽  
Bjørn Kjos-Hanssen
Keyword(s):  
Distributive Lattice ◽  
Representation Theorem ◽  
Initial Segment ◽  
Finite Lattice ◽  
The Other ◽  
Locally Finite ◽  
Other Hand ◽  
Lattice Representation ◽  
Quantifier Theory

AbstractWe affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, . In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable, locally finite lattice) is isomorphic to an initial segment of . Corollaries include the decidability of the two quantifier theory of , and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of ω1ck. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ω1. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of .

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.

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