Congruence Identity Research Articles

A q-Series Congruence Inspired by Andrews and Ramanujan

For each s∈{1,3,5}, we consider Rs(n) to be the number of the partitions of n into parts not congruent to 0, ±s(mod12). In recent years, some relations for computing the value of R3(n) were studied. In this paper, we investigate the parity of Rs(n) when s∈{1,5} and derive the following congruence identity: ∑n=1∞(−q;q)n−12(1+qn)qn2(q;q)2n≡∑n=1∞qn2+q3n2(mod2). For each s∈{1,5}, the number of the partitions of n into parts not congruent to 0, ±s(mod12) is connected with two truncated theta series. Some open problems involving R1(n) and R5(n) are introduced in this context.

Another characterization of congruence distributive varieties

AbstractWe provide a Maltsev characterization of congruence distributive varieties by showing that a variety 𝓥 is congruence distributive if and only if the congruence identity … (k factors) holds in 𝓥, for some natural number k.

Klein-Beltrami model. Part III

Summary Timothy Makarios (with Isabelle/HOL1) and John Harrison (with HOL-Light2) shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [2],[3],[4],[5]. With the Mizar system [1] we use some ideas taken from Tim Makarios’s MSc thesis [10] to formalize some definitions (like the absolute) and lemmas necessary for the verification of the independence of the parallel postulate. In this article we prove that our constructed model (we prefer “Beltrami-Klein” name over “Klein-Beltrami”, which can be seen in the naming convention for Mizar functors, and even MML identifiers) satisfies the congruence symmetry, the congruence equivalence relation, and the congruence identity axioms formulated by Tarski (and formalized in Mizar as described briefly in [8]).

The Jónsson distributivity spectrum

Suppose throughout that $\mathcal V$ is a congruence distributive variety. If $m \geq 1$, let $ J _{ \mathcal V} (m) $ be the smallest natural number $k$ such that the congruence identity $\alpha ( \beta \circ \gamma \circ \beta \dots ) \subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta \circ \dots $ holds in $\mathcal V$, with $m$ occurrences of $ \circ$ on the left and $k$ occurrences of $\circ$ on the right. We show that if $ J _{ \mathcal V} (m) =k$, then $ J _{ \mathcal V} (m \ell ) \leq k \ell $, for every natural number $\ell$. The key to the proof is an identity which, through a variety, is equivalent to the above congruence identity, but involves also reflexive and admissible relations. If $ J _{ \mathcal V} (1)=2 $, that is, $\mathcal V$ is $3$-distributive, then $ J _{ \mathcal V} (m) \leq m $, for every $m \geq 3$ (actually, a more general result is presented which holds even in nondistributive varieties). If $\mathcal V$ is $m$-modular, that is, congruence modularity of $\mathcal V$ is witnessed by $m+1$ Day terms, then $ J _{ \mathcal V} (2) \leq J _{ \mathcal V} (1) + 2m^2-2m -1 $. Various problems are stated at various places.

Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

We provide a partial result on Taylor's modularity conjecture, and several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and we prove an analogue for idempotent varieties with a cube term. Also, similar results are proved for linear varieties and the properties of congruence modularity, having a cube term, congruence $n$-permutability for a fixed $n$, and satisfying a non-trivial congruence identity.

Tarski Geometry Axioms – Part II

Summary In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation), of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1), congruence equivalence relation (A2), congruence identity (A3), segment construction (A4), SAS (A5), betweenness identity (A6), Pasch (A7). Also a simple model, which satisfies these axioms, was previously constructed, and described in [6]. In this paper, we deal with four remaining axioms, namely: the lower dimension axiom (A8), the upper dimension axiom (A9), the Euclid axiom (A10), the continuity axiom (A11). They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles cong is introduced via congruence of sides (SSS). In order to show that the structure which satisfies all eleven Tarski’s axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural [5] extension of ordinary metric structure Euclid 2 satisfies all these attributes. Although the tradition of the mechanization of Tarski’s geometry in Mizar is not as long as in Coq [11], first approaches to this topic were done in Mizar in 1990 [16] (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time [8]). Connection with another proof assistant should be mentioned – we had some doubts about the proof of the Euclid’s axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle [10] clarified things a bit. Our development allows for the future faithful mechanization of [13] and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory [7].

An easy test for congruence modularity

We describe an easy way to determine whether the realization of a set of idempotent identities guarantees congruence modularity or the satisfaction of a nontrivial congruence identity. Our results yield slight strengthenings of Day’s Theorem and Gumm’s Theorem, which each characterize congruence modularity.

Axiomatizable and nonaxiomatizable congruence prevarieties

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

Every $m$-permutable variety satisfies the congruence identity $\alpha\beta_h= \alpha \gamma_h$

It is known that congruence lattices of algebras in $ m$-permutable varieties satisfy non-trivial identities; however, the identities discovered so far are rather artificial and seem to have little intrinsic interest. We show here that every $ m$-permutable variety satisfies the well-known and well-studied congruence identity $ \alpha \beta_h= \alpha \gamma_h$. We also get a new condition equivalent to $ m$-permutability.

Person—environment congruence, self‐efficacy, and environmental identity in relation to job satisfaction: a career decision theory perspective

This study explored the relationship between person—environment congruence, self‐efficacy, and environmental identity and job satisfaction. Participants were 198 employees of a multinational telecommunications corporation. The predictor domain included the Iachan Index (R. Iachan, 1984), the Mahalanobis Distance Index (L. J. Cronbach & G. C. Gleser, 1953), the Self‐Efficacy Scale (M. Sherer et al., 1982, 2000), and the Environmental Identity Scale (G. D. Gottfredson & J. L. Holland, 1996; J. L. Holland, 1997). The criterion domain included 6 components of job satisfaction. A canonical correlation analysis identified 2 significant roots labeled organizational mission satisfaction and work task satisfaction. Implications for career decision making are discussed.

Chief Factor Sizes in Finitely Generated Varieties

AbstractLet A be a k-element algebra whose chief factor size is c. We show that if B is in the variety generated by A, then any abelian chief factor of B that is not strongly abelian has size at most ck−1. This solves Problem 5 of The Structure of Finite Algebras, by D. Hobby and R. McKenzie. We refine this bound to c in the situation where the variety generated by A omits type 1. As a generalization, we bound the size of multitraces of types 1, 2, and 3 by extending coordinatization theory. Finally, we exhibit some examples of bad behavior, even in varieties satisfying a congruence identity.

Congruence join semidistributivity is equivalent to a congruence identity

We show that a locally finite variety is congruence join semidistributive if and only if it satisfies a congruence identity that is strong enough to force join semidistributivity in any lattice.

A characterization of locally finite varieties that satisfy a nontrivial congruence identity

We show that a locally finite variety satisfies a nontrivial congruence identity if and only if it satisfies an idempotent Mal'tsev condition that fails in the variety of semilattices.

A modification of Polin's variety

In this note we settle a question posed by Hobby and McKenzie in [2] on the nature of locally finite equational classes which satisfy some nontrivial congruence identity.

Congruence Identities and their Local Versions

If an algebra A with a nullary operation c satisfies a congruence identity p=q, then it also satisfies [c]p=[c]q, the so-called identity at c. The converse is valid if A is weakly regular and the terms p, q do not contain the operation of relational product. We study these problems for 3-permutability and the so-called permutability at c.

Congruence semimodularity and identities

Let ~ be a variety. If an equation e in the language of lattices holds in Con(A) for all A ~ ~ , then we write C o n ( ~ ) 1= e and we say that ~ is a eongrneaee identity of ~e'. In the early 1970's it was conjectured that any nontrivial lattice identity holding as a congruence identity for a variety would imply congruence modularity, but the conjecture turned out to be false [6], [2]. In other words, there are nonmodular varieties having nontrivial congruence identities. The question that we examine in this note is: are there nonmodular congruence semimodular varieties having nontrivial congruence identities? We will show that the answer is negative, at least in the locally finite case. We recall that a lattice L is semimodular when, for all x, y, z E L, if x ^ y ~( x then x v y >-y. A variety C is congruence semimodular (CSM) if any algebra in ~/ has a semimodular congruence lattice. The first lemma we need is purely lattice theoretic and its proof can be found in [5] or [7].

Congruence and vocational identity: Outcomes of career counseling with persuasive power.

Congruence and vocational identity: Outcomes of career counseling with persuasive power.