In 1916 Ramanujan observed a remarkable congruence: τ(n) ≡ σ11(n) mod 691. The modern point of view is to interpret the Ramanujan congruence as a congruence between the Fourier coefficients of the unique normalized cusp form of weight 12 and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number B12. In this paper we give a simple proof of the Ramanujan congruence...

If 91 and 23 are algebras, where 9Is23, and 6 is a congruence relation on 91, let 0® be the smallest congruence relation on 93 containing 0. 91 is called a congruence subalgebra of 93 if 9ls93 and, for every congruence relation 0 on 91, 0*n|9I|2=0. Elementary subalgebras are congruence subalgebras, and there are Directed Union and Loewenheim-Skolem Theorems for congruence subalgebras analogous ...

This article develops and tests an algorithm of spatial congruence based on geometric congruity of two spatial areal objects in the Euclidean plane. Spatial congruence is defined and thus evaluated as an increasing continuous function of congruity in the position, orientation, size, and shape of spatial objects, dependent upon scaling, translation, and rotation. Expansion-based geometric matchi...

A systematic approach is developed for enumerating congruence classes of 2-cell imbeddings of connected graphs on closed orientable 2-manifolds. The method is applied to the wheel graphs and to the complete graphs. Congruence class genus polynomials and congruence class imbedding polynomials are introduced, to summarize important information refining the enumeration.

If V is a variety of algebras, let L(V) denote the prevariety of all lattices embeddable in congruence lattices of algebras in V . We give some criteria for the first-order axiomatizability or nonaxiomatizability of L(V). One corollary to our results is a nonconstructive proof that every congruence n-permutable variety satisfies a nontrivial congruence identity.

in this note, we introduce the concept of a fuzzy filter of a blalgebra,with respect to a t-norm briefly, t-fuzzy filters, and give some relatedresults. in particular, we prove representation theorem in bl-algebras. thenwe generalize the notion of a fuzzy congruence (in a bl-algebra) was definedby lianzhen et al. to a new fuzzy congruence, specially with respect to a tnorm.we prove that there i...

in this paper, we define the concepts of a complex fuzzy subset and a complex fuzzy finite state automaton. then we extend the notion of a complex fuzzy finite state automaton and introduce the notion of a general complex fuzzy automaton. after that we define the concept of a max- min general complex fuzzy automaton and construct some equivalence relations and some congruence relations in a max...

We define a G-fuzzy congruence, which is a generalized fuzzy congruence, discuss some of its basic properties, and characterize the G-fuzzy congruence generated by a fuzzy relation on a semigroup. We also give certain lattice theoretic properties of Gfuzzy congruences on semigroups.