[9] **viXra:1212.0157 [pdf]**
*submitted on 2012-12-27 16:40:31*

**Authors:** Colin Naturman

**Comments:** 20 Pages.

Certain topological countability properties are generalized to interior algebras and basic reaults concerning these properies are investigated. The preservation of these properties under the formation of principal quotients and under a new construction called a join of interior algebras, is investigated.

**Category:** Algebra

[8] **viXra:1212.0151 [pdf]**
*submitted on 2012-12-27 03:05:04*

**Authors:** Colin Naturman

**Comments:** 26 Pages. Results published in "Naturman, C.A., 1991, Interior Algebras and Topology, PhD Thesis, University of Cape Town Department of Mathematics" and this paper presented as "Research Reports, Department of Mathematics. The University of Cape Town, Volume 139

Interior algebras are Boolean algebras enriched with an interior operator and corresponding closure operator. Alternative descriptions of interior algebras in terms of generalized topologies in Boolean algebras and neighbourhood functions on Boolean algebras are found. The topological concepts of convergence and accumulation of systems and nets are generalized to interior algebras. Relationships between different forms of convergence and accunulation are found.

**Category:** Algebra

[7] **viXra:1212.0148 [pdf]**
*submitted on 2012-12-26 04:38:39*

**Authors:** Colin Naturman

**Comments:** 17 Pages. Results published in "Naturman, C.A., 1991, Interior Algebras and Topology, PhD Thesis, University of Cape Town Department of Mathematics" and this paper presented as "Research Reports, Department of Mathematics. The University of Cape Town, Volume 144

The intervals in an interior algebra A can be turned into interior algebras called interval algebras. Generalizations of homomorphisms, called topomorphisms, are introduced and certain quotient structures of A in the category of interior algebras and topomorphisms (the principal quotients) are shown to be (up to isomorphism) precisely the interval algebras of A.

**Category:** Algebra

[6] **viXra:1212.0146 [pdf]**
*submitted on 2012-12-25 11:02:30*

**Authors:** Peter Bruyns, Colin Naturman, Henry Rose

**Comments:** 17 Pages.

The amalgamation class Amal (N) of a lattice variety generated by a pentagon is considered. It is shown that Amal (N) is closed under reduced products and therefore is an elementary class determined by Horn sentences. The above result is based on a new characterization of Amal (N). The lattice varieties whose amalgamation classes contain Amal (N) as a subclass are considered.

**Category:** Algebra

[5] **viXra:1212.0138 [pdf]**
*submitted on 2012-12-23 08:11:59*

**Authors:** Colin Naturman, Henry Rose

**Comments:** 26 Pages.

The elementary equivalence of two full relation algebras, partition lattices or function monoids are shown to be equivalent to the second order equivalence of the cardinalities of the corresponding sets. This is shown to be related to elementary equivalence of permutation groups and ordinals. Infinite function monoids are shown to be ultrauniversal.

**Category:** Algebra

[4] **viXra:1212.0133 [pdf]**
*submitted on 2012-12-21 12:55:50*

**Authors:** Colin Naturman, Henry Rose

**Comments:** 25 Pages. Published in Journal of the Korean Mathematical Society, 30(1), 1993, pp.1–23 content is free for download but PDFs distributed by the publisher are missing the diagrams and/or abstract and errata.

An interior algebra is a Boolean algebra enriched with an interior operator. Congruences on interior algebras are investigated. Simple, subdirectly irreducible, finitely subdirectly irreducible and directly indecomposable interior algebras are characterized and the classes of these are shown to be finitely axiomatizable elementary classes. Quotients by open elements, dissectable and openly decomposable interior algebras are investigated. Basic results concerning interior algebras and their connection to topology are discussed.

**Category:** Algebra

[3] **viXra:1212.0131 [pdf]**
*submitted on 2012-12-21 03:39:27*

**Authors:** Colin Naturman, Henry Rose

**Comments:** 6 Pages. Published in Ordered Set and Lattices, 11, 1995 pp. 39-44, publisher does not provide offprints

An interior algebra is a Boolean algebra enriched with an interior operator. Given an interior algebra there is a natural way of forming interior algebras from its principal ideals. Basic results concerning these ideal algebras, Stone spaces of ideal algebras and preservation properties of ideal algebras are investigated.

**Category:** Algebra

[2] **viXra:1212.0124 [pdf]**
*replaced on 2012-12-21 04:13:24*

**Authors:** Deepak Ponvel Chermakani

**Comments:** There are 6 Pages, 6 Theorems, 7 Figures. I also made a small correction that in Theorem-1, the correct word is "NP-Hard" and not "NP-Complete".

We convert, within polynomial-time and sequential processing, an NP-Complete Problem into a real variable problem of minimizing a sum of Rational Linear Functions constrained by an Asymptotic-Linear-Program. The coefficients and constants in the real-variable problem are 0, 1, -1, K, or -K, where K is the time parameter that tends to positive infinity. The number of variables, constraints, and rational linear functions in the objective, of the real-variable problem is bounded by a polynomial function of the size of the NP-Complete Problem. The NP-Complete Problem has a feasible solution, if-and-only-if, the real-variable problem has a feasible optimal objective equal to zero. We thus show the strong NP-hardness of this real-variable optimization problem.

**Category:** Algebra

[1] **viXra:1212.0018 [pdf]**
*submitted on 2012-12-03 12:30:39*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 200 Pages.

The authors introduce the concept of neutrosophic super matrices and the new notion of quasi super matrices. This new notion of quasi super matrices contains the class of super matrices. The larger class contains more partitions of the usual simple matrices. Studies in this direction are interesting and find more applications in fuzzy models. The authors also suggest in this book some open problems.

**Category:** Algebra