Isotone Maps as Maps of Congruences
نویسنده
چکیده
Let L be a lattice and let L1, L2 be sublattices of L. Let be a congruence relation of L1. We extend to L by taking the smallest congruence of L containing . Then we restrict to L2, obtaining the congruence L2 of L2. Thus we have de ned a map ConL1 ! ConL2. Obviously, this is an isotone 0-preservingmap of the nite distributive lattice ConL1 into the nite distributive lattice ConL2. The main result of this paper is the converse. Let D1 and D2 be nite distributive lattices, and let : D1 ! D2 be an isotone map that preserves 0. Then there is a nite lattice L with sublattices L1 and L2 such that ConL1 represents D1 and ConL2 represents D2, and the map ConL1 ! ConL2 obtained by rst extending each congruence relation of L1 to L by minimal extension and then restricting the resulting congruence relation to L2 represents .
منابع مشابه
A Note on Euclidean Order Types
1. Ordered Classes, Isotone Maps, and Length Functions 1 1.1. Isotone Maps and Artinian Ordered Classes 1 1.2. Length Functions 2 2. Euclidean Functions 3 2.1. Basic Definitions 3 2.2. Structure Theory of Principal Rings 4 2.3. Generalized Euclidean Functions 4 2.4. Isotone Euclidean Functions 5 2.5. The Bottom Euclidean Function and the Euclidean Order Type 6 2.6. The Localized Euclidean Funct...
متن کاملIsotone maps on lattices
Let L = (Li | i ∈ I) be a family of lattices in a nontrivial lattice variety V, and let φi : Li → M , for i ∈ I, be isotone maps (not assumed to be lattice homomorphisms) to a common lattice M (not assumed to lie in V). We show that the maps φi can be extended to an isotone map φ : L → M , where L = FreeV L is the free product of the Li in V. This was known for L = V, the variety of all lattice...
متن کاملFuzzy order congruences on fuzzy posets
Fuzzy order congruences play an important role in studying the categoricalproperties of fuzzy posets. In this paper, the correspondence between the fuzzyorder congruences and the fuzzy order-preserving maps is discussed. We focus onthe characterization of fuzzy order congruences on the fuzzy poset in terms ofthe fuzzy preorders containing the fuzzy partial order. At last, fuzzy completecongruen...
متن کاملSome Results on Counting Linearizations of Posets
In §1 we consider a 3-tuple S = (|S|,4, E) where |S| is a finite set, 4 a partial ordering on |S|, and E a set of unordered pairs of distinct members of |S|, and study, as a function of n ≥ 0, the number of maps φ : |S| → {1, . . . , n} which are both isotone with respect to the ordering 4, and have the property that φ(x) 6= φ(y) whenever {x, y} ∈ E. We obtain results on this function, which we...
متن کاملMULTIVARIATE p-ADIC FORMAL CONGRUENCES AND INTEGRALITY OF TAYLOR COEFFICIENTS OF MIRROR MAPS
We generalise Dwork’s theory of p-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely, with z = (z1, z2, . . . , zd), we show that the Taylor coefficients of the multi-variable series q(z) = zi exp(G(z)/F (z)) are integers, where F (z) and G(z) + lo...
متن کامل