Monoid varieties with extreme properties

Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids that generate monoid varieties with continuum many subvarieties; these include any finite inherently non-finitely based monoid and any monoid for which $xyxy$ is an isoterm. It follows that the join of two Cross monoid varieties can have a continuum cardinality subvariety lattice that violates the ascending chain condition. Regarding monoid varieties with countably infinitely many subvarieties, the first example of a finite monoid that generates such a variety is exhibited. A complete description of the subvariety lattice of this variety is given. This lattice has width three and contains only finitely based varieties, all except two of which are Cross

The Art of Learning Community: Technology and Gamification As A Recipe For Learning Umami

How can arts, technology, mobile apps, and a learning community be used as flavors to create an engaging learning platform to motivate students?School reform efforts purport to create engaged learners that can think creatively beyond the standards, however, teachers struggle with how to reconcile the culture of standardized testing with the learner engagement and motivation that is key to student success. When designing learning experiences that promote creativity, via information and computer technology, teachers need to adopt an ecological approach that encompasses people, practices, values, and technology interacting- with the spotlight being on human activities. The Japanese word umami describes how humans engage all senses to form judgments about their food. This provides an apt metaphor for instructional design. Food should be nourishing, presentable, and delicious- a feast for the senses. This is a worthy standard for any lesson- the goal of “learning umami.” The author proposed to create an analogous process in crafting an online learning community (http://edvislee.wix.com/rehearse-for-life), which consists of a mash-up of tools, apps, content, gamification, and collaboration with artists as “flavors” for engagement. This paper will review the community’s features before and after modifications, discuss design implications and rationale for changes, and make recommendations for additional improvements. The results demonstrate how pedagogy, design, and evaluation can be used to tailor existing apps, tools, services, and content to create a compelling learning community to meet any instructional design challenge

Chinese parents\u27 support for the bilingual educational program

Purpose: The purpose of this study was to examine support for the bilingual education program. Specifically, this study focused on parents whose children are in bilingual education classes in the elementary grades and sought to explore the relationship between support and several independent variables. They were: (a) socioeconomic status of the parents, (b) parental involvement with the program, and (c) parent influence in the program. Procedure: Questionnaires were sent to 256 Chinese parents who had children in an elementary bilingual education program in Oakland. A total of 191 or 76.4 percent returned the survey. The respondents were asked to respond to questions. The questionnaire was divided into three sections consisting of questions designed to provide information about the following areas: (1) socioeconomic status; (2) parents\u27 involvement; and (3) parent influence in the program. The data were computer processed using the Statistical Package for the Social Science. Findings: Three null hypotheses were tested. Hypothesis one stated that there is no relationship between level of support for the bilingual program and parent socioeconomic status. The study found no significant statistical difference between parent support and socioeconomic status. However, a further analysis of income indicated a negative relationship to parental support. Hypothesis two stated that there is no relationship between level of support for the bilingual education program and parent involvement. The findings reveal that parent involvement is correlated with parent support in a positive manner. Hypothesis two is rejected. Hypothesis three stated that there is no relationship between level of support for the bilingual education program and parent influence in the program. The findings reveal that parent influence is not correlated with parent support. Hypothesis is retained. Recommendations: Additional research is recommended in four areas: 1) A study to clear up conceptually the two bilingual program terms, maintenance and transition. 2) A study of recent immigrant parents from different ethnic groups to see why or if they want bilingual education. 3) A study to compare immigrant families in order to ascertain if there is a trend for them to become less supportive of bilingual education as they become more economically successful. 4) An interview methodology to be done with a larger and more economically diverse population, which might yield greater understanding of these issues

Combinatorial Rees–Sushkevich Varieties That Are Cross, Finitely Generated, Or Small

A variety is said to be a Rees–Sushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. Recently, all combinatorial Rees–Sushkevich varieties have been shown to be finitely based. The present paper continues the investigation of these varieties by describing those that are Cross, finitely generated, or small. It is shown that within the lattice of combinatorial Rees–Sushkevich varieties, the set ℱ of finitely generated varieties constitutes an incomplete sublattice and the set � of small varieties constitutes a strict incomplete sublattice of ℱ. Consequently, a combinatorial Rees–Sushkevich variety is small if and only if it is Cross. An algorithm is also presented that decides if an arbitrarily given finite set Σ of identities defines, within the largest combinatorial Rees–Sushkevich variety, a subvariety that is finitely generated or small. This algorithm has complexity �(nk) where n is the number of identities in Σ and k is the length of the longest word in Σ

Finitely Based Monoids Obtained From Non-Finitely Based Semigroups

Presently, no example of non-finitely based finite semigroup S is known for which the monoid S1 is finitely based. Based on a general result of M. V. Volkov, two methods are established from which examples of such semigroups can be constructed

On the Complete Join of Permutative Combinatorial Rees–Sushkevich Varieties

A semigroup variety is a Rees–Sushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. The collection of all permutative combinatorial Rees–Sushkevich varieties constitutes an incomplete lattice that does not contain the complete join J of all its varieties. The objective of this article is to investigate the subvarieties of J. It is shown that J is locally finite, non-finitely generated, and contains only finitely based subvarieties. The subvarieties of J are precisely the combinatorial Rees–Sushkevich varieties that do not contain a certain semigroup of order four

Finite Basis Problem for 2-Testable Monoids

A monoid S 1 obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S 1 is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based

Maximal Specht Varieties of Monoids

A variety of algebras is a Specht variety if all its subvarieties are finitely based. This article presents the first example of a maximal Specht variety of monoids. The existence of such an example is counterintuitive since it is long known that maximal Specht varieties of semigroups do not exist. This example permits a characterization of Specht varieties in the following four classes based on identities that they must satisfy and varieties that they cannot contain: (1) overcommutative varieties, (2) varieties containing a certain monoid of order seven, (3) varieties of aperiodic monoids with central idempotents, and (4) subvarieties of the variety generated by the Brandt monoid of order six. Other results, including the uniqueness or nonexistence of limit varieties within the aforementioned four classes, are also deduced. Specifically, overcommutative limit varieties of monoids do not exist. In contrast, the limit variety of semigroups, discovered by M.V. Volkov in the 1980s, is an overcommutative variety

Maximal Clifford Semigroups of Matrices

All maximal Clifford semigroups of matrices are identified up to isomorphism. If the ground field of the matrices is finite, then there exists a unique Clifford semigroup of maximum order

Finite Involution Semigroups with Infinite Irredundant Bases of Identities

A basis of identities for an algebra is irredundant if each of its proper subsets fails to be a basis for the algebra. The first known examples of finite involution semigroups with infinite irredundant bases are exhibited. These involution semigroups satisfy several counterintuitive properties: their semigroup reducts do not have irredundant bases, they share reducts with some other finitely based involution semigroups, and they are direct products of finitely based involution semigroups