Solution to a problem of FitzGerald

FitzGerald identified four conditions (RI), (UR), (RI*) and (UR*) that are necessarily satisfied by an algebra if its monoid of endomorphisms has commuting idempotents. We show that these conditions are not sufficient, by giving an example of an algebra satisfying the four properties, such that its monoid of endomorphisms does not have commuting idempotents. This settles a problem presented by FitzGerald at the Conference and Workshop on General Algebra and Its Applications in 2013 and more recently at the workshop NCS 2018. After giving the counterexample, we show that the properties (UR), (RI*) and (UR*) depend only on the monoid of endomorphisms of the algebra, and that the counterexample we gave is in some sense the easiest possible. Finally, we list some categories in which FitzGerald’s question has an affirmative answer.

All idempotent and regular elements in the monoid of generalized hypersubstitutions for algebraic systems of type (2; 2)

There are two different concepts for hypersubstitutions for algebraic systems [K. Denecke and D. Phusanga, Hyperformulas and solid algebraic systems, Studia Logica 90(2) (2008) 263–286; J. Koppitz and D. Phusanga, The monoid of hypersubstitutions for algebraic systems, J. Announcements Union Sci. Sliven 33(1) (2018) 120–127]. In this paper, we follow the more natural and practicable one given in [J. Koppitz and D. Phusanga, The monoid of hypersubstitutions for algebraic systems, J. Announcements Union Sci. Sliven 33(1) (2018) 120–127]. On the other hand, in [S. Leeratanavalee and K. Denecke, Generalized hypersubstitutions and strongly solid varieties, General Algebra and Applications[Formula: see text] Proc. of 59th Workshop on General Algebra[Formula: see text] 15th Conf. for Young Algebraists Potsdam 2000 (Shaker Verlag, 2000), pp. 135–145], the concept of the monoid of generalized hypersubstitutions was introduced. Following both ideas, one obtains the concept of a monoid of generalized hypersubstitutions for algebraic systems in a canonical way. The purpose of this paper is the study of the monoid of generalized hypersubstitutions for algebraic systems. We characterize the idempotent as well as regular elements in this monoid.

Importance of Grades and Placement for Math Attainment

Research on high school math course taking documents the advantages of starting high school at or beyond Algebra 1. Fewer studies examine differentiation into remedial, general, and honors Algebra 1 course types by course rigor. This study examines how course grades and course rigor are associated with math attainment among students with similar eighth-grade standardized math test scores. Students who earned an A in remedial courses had lower attainment than students with a D in general Algebra 1. Students with an A in general Algebra 1 had lower attainment than students with median grades in honors Algebra 1.

Hypersatisfaction of quantifier free formulas in algebraic systems

In [Hyperformulas and solid algebraic systems, Studia Logica 90(2) (2008) 263–286], the theory of hyperidentities and solid varieties (see [K. Denecke, D. Lau, R. Pöschel and D. Schweigert, Hyperidentities, hyperequational classes, and clone congruences, in Contributions to General Algebra, Vol. 7 (Verlag Hölder-Pichler-Tempsky, Wien, 1991), pp. 97–118]) was extended to algebraic systems and solid model classes of algebraic system. In this paper, we will present a different approach which is based on the concept of the term operations and the realization of quantifier free formulas.

On the Definitional Embeddability of Some Elementary Algebraic Theories into the First-Order Predicate Calculus

In this article we prove a theorem on the definitional embeddability into first-order predicate logic without equality of such well-known mathematical theories as group theory and the theory of Abelian groups. This result may seem surprising, since it is generally believed that these theories have a non-logical content. It turns out that the central theory of general algebra are purely logical. Could this be the reason that we find them in many branches of mathematics? This result will be of interest not only for logicians and mathematicians but also for philosophers who study foundations of logic and its relation to mathematics.

Soft Set Theory Applied to General Algebras

The notions of a soft general algebra and a soft subalgebra are introduced and studied. The operations on them such as a restricted intersection, an extended intersection, a restricted union, a ∧-intersection, a ∨-union and a cartesian product are established.