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We denote by Conc(A) the semilattice of compact congruences of an algebra A. Given a variety V of algebras, we denote by Conc(V) the class of all semilattices isomorphic to Conc(A) for some A in V. Given varieties V1 and V2 varieties of algebras, the critical point of V1 under V2, denote by crit(V1;V2) is the smalest cardinality of a semilattice in Conc(V1) but not in Conc(V2). Given a finitely generated variety V of modular lattices, we obtain an integer l, depending of V, such that crit(V;Var(Sub F^n)) is at least aleph_2 for any n > 1 and any field F. In a second part, we prove that crit(Var(Mn);Var(Sub F^3))=aleph_2, for any finite field F and any integer n such that 1+card F< n. Similarly crit(Var(Sub F^3);Var(Sub K^3))=aleph_2, for all finite fields F and K such that card F>card K.
We denote by Conc(L) the semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor its dual, and such that every simple member of W contain
We denote by Conc(A) the semilattice of all finitely generated congruences of an (universal) algebra A, and we define Conc(V) as the class of all isomorphic copies of all Conc(A), for A in V, for any variety V of algebras. Let V and W be locally fini
For a class V of algebras, denote by Conc(V) the class of all semilattices isomorphic to the semilattice Conc(A) of all compact congruences of A, for some A in V. For classes V1 and V2 of algebras, we denote by crit(V1,V2) the smallest cardinality of
In this paper we show how to approximate (learn) a function f, where X and Y are metric spaces.
Ultrafilters are useful mathematical objects having applications in nonstandard analysis, Ramsey theory, Boolean algebra, topology, and other areas of mathematics. In this note, we provide a categorical construction of ultrafilters in terms of the in