Congruence Lattices of Ideals in Categories and (Partial) Semigroups

We prove that a group G is Abelian whenever (1) it is nilpotent and the lattice of normal subgroups of G is isomorphic to the subgroup lattice of an Abelian group or (2) there exists a non-torsion Abelian group B such that the normal subgroup lattice of B × G is isomorphic to the subgroup lattice of an Abelian group. Using (2), it is proved that an Abelian group A can be determined in the class of all groups by the lattice of all normal subgroups of some groups, e.g. if A is an Abelian group and G is a group such that \mathbb Z × A and \mathbb Z × G have isomorphic normal subgroup lattices then A and G are isomorphic groups.

We study D-hyperoctahedral groups, diagonal inductive limits of hyperoctahedral groups. Also, we describe the Z 2-modules of periodic sequences over diagonal limits of symmetric groups under their action on the elements of a Z 2-module by permutation of coordinates. The imprimitivity systems for D-hyperoctahedral groups are characterized, and a full description of the lattices of their normal subgroups is given.

In this paper, we discuss the sublattices of intuitionistic fuzzy subgroups of a given group and some special sublattices corresponding to it. We prove that a class of intuitionistic fuzzy normal subgroups constitutes a modular of the lattice of intuitionistic fuzzy normal subgroups.

One way of expressing the self-duality $A\cong {\rm Hom}(A,\mathbb{C})$ of Abelian groups is that their character tables are self-transpose (in a suitable ordering). In this paper we extend the duality to some noncommutative groups considering when the character table of a finite group is close to being the transpose of the character table for some other group. We find that groups dual to each other have dual normal subgroup lattices. We show that our concept of duality cannot work for non-nilpotent groups and we describe p-group examples.

For a given group G and a homomorphism ϕ: G → G × G, we construct groups ℱϕ(G), 𝒯ϕ(G), and 𝒱ϕ(G) that blend Thompson's groups F, T, and V with G, respectively. Furthermore, we describe the lattice of normal subgroups of the groups ℱΔ(G), where Δ: G → G × G is the diagonal homomorphism, Δ(g) = (g, g).

We introduce and study the lattice of normal subgroups of a group G that determine solitary quotients. It is closely connected to the well-known lattice of solitary subgroups of G, see [Kaplan G., Levy D., Solitary subgroups, Comm. Algebra, 2009, 37(6), 1873–1883]. A precise description of this lattice is given for some particular classes of finite groups.

Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals

We correct an error in [2] by exhibiting another ideal in the lattice of ideals when the normal subgroup used in the construction of the nearring has order two.

A subgroup H of a group G is said to be nearly normal in G if it has a finite index in its normal closure H G.T he setnn(G) of nearly normal subgroups of G is a sublattice of the lattice of all subgroups of G. Isomorphisms between lattices of nearly normal subgroups of FC -soluble groups are considered in this paper. In particular, properties of images of normal subgroups under such an isomorphism are investigated. Moreover, it is proved that if G is a supersoluble group and ¯ G is an FC -soluble group such that the lattices nn(G )a ndnn( ¯ G )a re isomorphic, then also ¯ G is supersoluble.

The objective of this transformative action research project was to explore and develop sustainable methods to promote female empowerment through science education in rural, disadvantaged sectors of South Africa. In an attempt to achieve this we collaborated with local community members to develop and implement a contextualized science curriculum at a school in the aforementioned setting. As soon as the project was launched it became increasingly clear that although the ideology of ‘empowerment through science education’ seemed a promising venture, it could also be an extremely complex and often frustrating undertaking. This was especially true when working within an unfamiliar cultural setting. Numerous challenges, such as the lack of teacher motivation, malnutrition amongst the learners, and conflicts stemming from differences between the indigenous knowledge and the western concept of science, greatly impeded the delivery of quality education in the area. These challenges had to be addressed both in pedagogical and practical terms before any attempt towards libratory education could be made. This article sheds light on the complex inter-relationship between the human factor and the organizational and physical infrastructure at a school. It begins with a brief description of the local context and goes on to identify the theoretical underpinnings and chosen methodology for the project. The article concludes with a review of the complexities involved in possible attempts to initiate and foster educational and social transformations in a rural South African setting. We contend that it is essential to first be thoroughly familiar with the background, culture, and needs of any community before any attempts are made towards social justice.

We discuss the relationship between interval-valued fuzzy ideals and interval-valued fuzzy congruence on a distributive lattice L and show that for a generalized Boolean algebra the lattice of interval-valued fuzzy ideals is isomorphic to the lattice of interval-valued fuzzy congruences. Finally we consider the products of interval-valued fuzzy ideals and obtain a necessary and sufficient condition for an interval-valued fuzzy ideal on the direct sum of lattices to be representable as a direct sum of interval-valued fuzzy ideals on each lattice.

In J6nsson [6] and Day [1] Mal'cev type characterizations were given for congruence distributivity and congruence modularity respectively. (See e.g. Taylor [9] for the precise definitions and the characterization of Mal'cev type conditions in general.) Gedeonovfi [5] and Mederly [7] showed that other (non-trivial) lattice identities were Mal'cev type conditions but Day [2], Nation [8] and [7] showed that these conditions imply (and hence are equivalent) to either congruence distributivity or congruence modularity. These facts (along with others to be stated below) have raised the following two conjectures (1) McKenzie: If the congruence lattices of a variety satisfy any non-trivial lattice identity, then the variety is congruence modular. (2) Nation: If a non-trivial lattice identity has a Mal'cev type characterization, then it implies congruence modularity. There are particular results in the case of the first conjecture. It has been proven true for varieties of unary algebras (Nation [8]), semigroups (Evans [3] and Freese and Nation [4]) and semilattices ([8]). The first general result, referring to lattice identities of a special form, was due to Nation in [8].

Corrigendum to “Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals” [J. Symb. Comput. 50 (March 2013) 314–334

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings, maximal rings, etc. Inspired by LIP, lifting properties were also defined for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congruence distributive universal algebras, etc. In this paper, we define a lifting property (LP) in commutative coherent integral quantales, structures that are a good abstraction for lattices of ideals, filters and congruences. LP generalizes all the lifting properties existing in the literature. The main tool in the study of LP in a quantale A is the reticulation of A, a bounded distributive lattice whose prime spectrum is homeomorphic to the prime spectrum of A. The principal results of the paper include a characterization theorem for quantales with LP and a characterization theorem for hyperarchimedean quantales.

We prove that the lattice of fuzzy congruences of a group G (resp. ring R) is isomorphic to the lattice of fuzzy normal subgroups of G (resp. fuzzy ideals of R).