Shelly L. Wismath

Galois Connections and Applications

Preface M. Erne Adjunctions and Galois Connections: Origins, History and Development G. Janelidze Categorical Galois Theory: Revision and Some Recent Developments M. Erne The Polarity between Approximation and Distribution K. Denecke, S.L. Wismath Galois Connections and Complete Sublattices R. Poschel Galois Connections for Operations and Relations K. Kaarli Galois Connections and Polynomial Completeness K. Glazek, St. Niwczyk Q-Independence and Weak Automorphisms A. Szendrei A Survey of Clones Closed Under Conjugation P. Burmeister Galois Connections for Partial Algebras K. Denecke, S.L. Wismath Complexity of Terms and the Galois Connection Id-Mod J. Lambek Iterated Galois Connections in Arithmetic and Linguistics I. Chajda, R. Halas Deductive Systems and Galois Connections J. Slapal A Galois Correspondence for Digital Topology W. Gahler Galois Connections in Category Theory, Topology and Logic R. Wille Dyadic Mathematics - Abstractions from Logical Thought Index

Hyperidentities and clones

Basic Concepts. Hyperidentities and Hypervarieties. Hyperidentities for Varieties of Semigroups. M-Hyperidentities and M-Solid Varieties. M-Solid Varieties of Semigroups. Describing Clones by Hyperidentities. Hyperidentities in Partial Algebras. Exercises and Problems.

Shifting the Teacher-Learner Paradigm: Teaching for the 21st Century

Designing and teaching a post-secondary elective Liberal Education course in “Puzzles and Problems” prompts a veteran mathematics professor to re-examine the roles of student and instructor in a university classroom. As students became authentically engaged in collaborative and cooperative development of problem-solving skills, and the classroom environment shifted from teacher-centric to learner-centric, the role of the “teacher” changed radically to a new teacher-learner paradigm, reflecting the emerging 21st century learning environment as it affects both teachers and students.

Complexity of terms, composition, and hypersubstitution

We consider four useful measures of the complexity of a term: the maximum depth (usually called the depth), the minimum depth, the variable count, and the operation count. For each of these, we produce a formula for the complexity of the composition Smn(s,t1,…,tn) in terms of the complexity of the inputs s, t1,…, tn. As a corollary, we also obtain formulas for the complexity of σˆ[t] in terms of the complexity of t when t is a compound term and σ is a hypersubstitution. We then apply these formulas to the theory of M-solid varieties, examining the k-normalization chains of a variety with respect to the four complexity measures.

Solid varieties of semigroups

An identityt≈t′ of terms of a given type is called a hyperidentity of a varietyV (not necessarily of the same type) if whenever the operation symbols occurring int andt′ are replaced by any terms ofV of the appropriate arity, the identity which results holds inV. A varietyV of type ϑ is called solid if every identity ofV also holds as a type ϑ hyperidentity inV. Denecke, Lau, Pöschel and Schweigert have shown that a varietyV is solid if and only ifV is a hyperequational class. We use this result, along with the equational description by Denecke and Koppitz of the hyperassociative variety of semigroups, to characterize solid semigroup varieties, and to produce some new examples of such varieties, including some infinite ascending chains in the lattice of solid semigroup varieties.

Hyperidentities for some varieties of commutative semigroups

Hyperidentities and hypervarieties have been defined by Taylor in [5]. A hypervariety is a class of varieties closed under the formation of equivalent, product, reduct and subvarieties. Hyperidentities are used to define hypervarieties, in the same way that ordinary identities define varieties. This paper produces some hyperidentities satisfied by various varieties of commutative semigroups, and identifies some restrictions as to what kind of hyperidentities such varieties can satisfy. It also continues the study, begun in [6], of the closure and hypervariety operators defined there, as they apply to varieties of commutative semigroups.

Externalization of lattices

Let r be a type of algebras. An identity s m t of type r is said to be externally compatible, or simply external, if the terms s and t are either the same variable or both start with the same operation symbol f j of the type. A variety is called external if all of its identities are external. For any variety V, there is a least external variety E(V) containing V, the variety determined by the set of all external identities of V. External identities and varieties have been studied by [4], [5] and [2], and a general characterization of the algebras in E(V) has been given in [3]. In this paper we study the algebras of the variety E(V) where V is the type (2,2) variety L of lattices. Algebras in L may also be described as ordered sets, and we give an ordered set description of the algebras in E(L). We show that on any algebra in E(L) there is a natural quasiorder having an additional property called externality, and that any set with such a quasiorder can be given the structure of an algebra in E(L). We also characterize algebras in E(L) by an inflation construction.

Hyperidentities for some varieties of semigroups

Hyperidentities and hypervarieties have been defined by Taylor in [4]. A hypervariety is a class of varieties closed under the formation of equivalent, product, reduct and sub-varieties. Hyperidentities are used to define hypervarieties, in the same way that ordinary identities define varieties. In this paper we consider hyperidentities for hypervarieties generated by two types of varieties of semigroups, varieties of bands and varieties of nilpotent semigroups. We introduce two operators on the lattice of varieties of semigroups, the closure and hypervariety operators, and study their properties.

BASES FOR THE k-NORMALIZATIONS OF VARIETIES OF BANDS

The usual depth measurement on terms of a fixed type type r assigns to each term a non-negative integer called its depth. For k > 1, an identity s « t of type r is said to be Abnormal (with respect to the depth measurement) if either s = t or both s and t have depth > k. Taking k = 1 gives the well-known property of normality of identities. A variety is called Abnormal (with respect to the depth measurement) if all its identities are A;-normal. For any variety V, there is a least fc-normal variety Nk(V) containing V, the variety determined by the set of all fc-normal identities of V. In this paper we produce for every subvariety V of the variety B of bands (idempotent semigroups) a finite equational basis for Nk(V), for k > 1.

A Characterization of P-Compatible Varieties

P-Compatibility is a hereditary property of identities which generalizes the properties of normality and externality of identities. Chajda characterized the normalization of a variety by an algebraic construction called a choice algebra. In this paper, we generalize this characterization to the least P-compatible variety P(V) determined by a variety V for any partition P using P-choice algebras. We also study the clone of (strongly) P-compatible n-ary terms of a variety V, and relate identities of this clone to (strongly) P-compatible hyperidentities of the variety V.