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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING

Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (INTERNATIONAL JOURNAL) UGC APPROVED NO. 48767, ISSN No. 2456-1037

Vol.02, Issue 03, March 2017, Available Online: www.ajeee.co.in/index.php/AJEEE

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STUDY FOR CATEGORIES OF MEROTOPIC AND ALGEBRAS

Dr. H. K. Tripathi

Lecturer, Govt. Women’s Polytechnic College, Jabalpur-482001

Abstract- We study arithmetical properties of classifications of Merotopic, Nearness, and Filter Algebras. We show that the classification of filter twist free abelian bunches is a perfective subcategory of the classification of filter abelian gatherings. The distracted functor from the classification of filter rings to filter monoids is basically logarithmic and the neglectful factor from the class of _lter gatherings to the class of filters has a left adjoint.

Keywords: Universal polynomial math, topological polynomial math, closeness spaces, merotopic spaces, lter spaces.

1 INTRODUCTION

We first depict three classes which contain Top, the classification of topological spaces (now and then with a partition maxim). They are Mer, the classification of the merotopic spaces of Katetov [8], Near, the classification of proximity spaces of Herrlich [4], and Fil, the class of filter spaces of Katetov [8].

Leave X alone a set and P2(X) be the set whose individuals are largely assortments of subsets of X. For any part An of P2(X), we compose

If are two merotopic, prenearness, or nearness spaces, then a mapping is called uniformly continuous

Ag is a member of for each

P-Near, Mer, and Near are the categories with objects which are prenearfiness, merotopic, and nearness spaces respectively with uniformly continuous mappings as morphisms. Mer is a bicore ective full subcategory of P- Near and Near is a bire ective full subcategory of Mer (see [5]).

If and are two

merotopic, prenearness, or nearness spaces, then a mapping is called uniformly continuous

is a member of for each A 2.

, and Near are the categories with objects which are prenearness, merotopic, and nearness spaces respectively with

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2 uniformly continuous mappings as morphisms. Mer is a bicoreective full subcategory of P-Near and Near is a bireective full subcategory of Mer. If X denotes any one of these categories, any X-object is simply denoted by X if the X-structure on X is understood. Moreover, a collection A of subsets of X is said to be

(1) Near in X provided A is a member of

(2) Micromeric in X provided the collection sec A is near in X, (3) Far in X provided A is not near

in X,

(4) Uniform cover of X provided the collection fXnA: A 2 Ag is far in X.

An is known as a stack on X i_ A = stack XA. The design of a merotopic space is controlled by the arrangement of merotopic stacks in light of the fact that an assortment is micromeric its stack in X is micromeric. A lter on X is supposed to be a Cauchy it is micromeric on X. X is known as a filter-merotopic (or simply a filter) space each micromeric stack contains a Cauchy filter. The full subcategory of Mer whose articles are all _lter spaces will be meant by Fil. This class Fil is cartesian shut (see [9]) and is bicoreective and innate in Mer.

A family of

natural numbers indexed by some set J is called a type. The index set J is called the order of . In the

following, we let a type be xed. A pair of a set jAj

and a family

of mappings is called an Ω-algebra (see, for example, [2]). For the sake of simplicity, we write A instead of

the pair for the

nj-ary operation Aj on A. If the algebra A is clear from the context, we drop the su x A in denoting its operation. If A and B are algebras, then a mapping is said to be an Ω- morphism iff for each

where n = nj

and is the mapping

with the obvious definition The symbol Alg(Ω) denotes the category whose objects are Ω- algebras and whose morphisms are Ω-morphisms.

Let X be a construct with nite concrete powers and A be a subcategory of Alg(Ω).

By a paired object (from X and A) is meant an ordered pair (X;A) where X and A are objects in X and A respectively with the same underlying set such that, for each j 2 J, the n(=nj)-ary operation on A is an X- morphism

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3 In this case, we write for the X-morphism from Xn to X whose

underlying function is

are two paired objects (from X and A), then an X- morphism that is also an A-morphism is called a paired morphism (from X and A) and is denoted by

The category of all paired objects (from X and A) together with paired morphisms (from X and A) is called the paired category (from X and A).

We denote this category by X⋄ A.

In this work, we assume that all subcategories are isomorphism closed. The fact that the most of the natural subcategories fall into these class justices our assumption.

Unless otherwise stated, X and Y denote arbitrary constructs with finite concrete powers, and A represents any subcategory of Alg(Ω). We write for the underlying set of an object X in a construct. For effortlessness, we will mean an item (X,A) in the matched classification (from X and A) either by X or by A. We will utilize a comparative identification for morphisms in the matched class.

2. BASICALLY LOGARITHMIC AND MATHEMATICAL

SUBCATEGORIES

We investigate logarithmic properties of matched classes with the accompanying lemma whose

confirmation can be found in [3].

Lemma 2.1 Suppose that X is mono topological, A will be a subcategory of Alg(Ω′);B is a basically arithmetical subcategory of Alg(ω), and is a concrete functor such that the association

Where T and T′ are absent minded functors. Then, at that point the accompanying hold.

(a) If H has a left adjoint, then, at that point ~H additionally has a left adjoint and ~H is (producing, monosource) - factorizable.

(b) If H reects isomorphisms, then, at that point ~H reects isomorphisms.

(c) If H is basically mathematical, then, at that point ~H is basically arithmetical.

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4 Proposition 2.2 If X is any one of the categories Mer, P-Near, Near, or Fil, then the forgetful functors X⋄Rng ! X⋄Ab and X⋄Rng ! X⋄Mon (Rng, Ab, and Mon are the categories of rings, abelian groups, or monoids repectively) are essentially algebraic.

Proof. Since the forgetful functors Rng ! Ab and Rng ! Mon are essentially algebraic, the associated forgetful functors are essentially algebraic by Lemma 2.1 (part (c)).

Lemma 2.3 Suppose that X is monotopological, A is a subcategory of Alg(Ω′), and B is an essentially algebraic subcategory of Alg(Ω).

(1) If B is also a reflective subcategory of A, then X⋄B is a reflective subcategory of X⋄A.

(2) If B is also an epireflective subcategory of A, then X⋄B is an flepireflective sub-category of X⋄A.

Proof. The inclusion map is essentially algebraic, and hence ~H, defined as in Lemma 2.1, is essentially algebraic (by Lemma 2.1). Thus is a reactive subcategory of . For the second part, if B is also an epireective subcategory of A, then has a left adjoint, and hence ~H has left adjoint (by Lemma 2.1), which in turn implies

that is an epireective subcategory of X A.

REFERENCES

1. J. Adamek, H. Herrlich and G. E.

Strecker, Abstract and Concrete Categories. John Wiley and Sons, Inc., New York, 1990.

2. P. M. Cohn, Universal Algebra. Harper and Row, Publishers, New York, 1965.

3. V. L. Gompa, Essentially logarithmic functors and topological polynomial math. Indian Journal of Mathematics, 35, (1993), 189-195.

4. H. Herrlich, An idea of proximity.

General Topology and its Applications 4 (1974), 191-212.

5. H. Herrlich, Topological designs. In:

Topological designs. Math. Focus Tracts 52 (1974), 59-122.

6. H. Herrlich, G. E. Strecker, Category Theory. Allyn and Bacon, Boston, 1973.

7. Y. H. Hong, Studies on classifications of widespread topological algebras.

Doctoral Dissertation, McMaster University, 1974.

8. M. Katetov, On progression designs and spaces of mappings. Remark.

Math. Univ. Tune. 6 (1965), 257-278.

9. M. Katetov, Convergence structures.

General Topology and its Applications II, Academic Press, New York (1967), 207-216.

10. O. Wyler, On the classifications of general geography and topological algebras. Bend. Math. (Basel) 22 (1971), 7-17.