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http://hdl.handle.net/123456789/178| Title: | ENUMERATION OF CERTAIN ALGEBRAIC SYSTEMS AND RELATED RESULTS |
| Authors: | Bhavale, Ashok N. Waphare, B.N. (Guide) |
| Keywords: | Mathematics ALGEBRAIC SYSTEMS Birkhoff’s open problems Structure theorem Nullity of a poset Ore’s open problem Whitney type characterization Enumeration of lattices Basic blocks Counting fundamental basic blocks Enumeration of blocks on six reducible elements Enumeration of blocks on five reducible elements Enumeration of blocks on four reducible elements |
| Issue Date: | Dec-2013 |
| Publisher: | UNIVERSITY OF PUNE |
| Description: | The theory of ordered sets is today a burgeoning branch of mathematics. It both draws upon and applies to several other branches of mathematics, including algebra, set theory, and combinatorics. The theory itself boasts an impressive body of fundamental and deep results as well as a variety of challenging problems, some of traditional heritage and some of fairly recent origin. Ordered sets have their roots in two trends of nineteenth century mathematics. On the one hand, ordered sets have entered into the study of those algebraic systems which originally arose from axiomatic schemes aimed at formalizing the “laws of thought”; Boole, Peirce, Schr¨ oder, and Huntington were among the earliest leaders of this trend. On the other hand, ordered sets were essential ingredients to the theory of sets, from its inception. It is not surprising that these two trends have influenced the subject in different ways. The ordered sets of most interest to general algebra are lattices. It is lattice theory, however, that has stimulated the study of ordered sets as abstract systems. The theory of lattices is bracketed under Universal Algebra, one of the major branches of Algebra. Orders are everywhere in mathematics and related fields like computer science. Partial order and lattice theory have applications in distributed computing, programming language semantics and data mining. Much of the combinatorial interest in ordered sets is inextricably linked to the combinatorial features of the diagrams associated with them. Ore[20] raised an open problem, namely, “Characterize those graphs which are orientable”. It is also well known that a graph G is the comparability graph of an ordered set if and only if each odd cycle of G has a triangular chord. In contrast little is known about this question : when is a graph the covering graph of an ordered set? Also, it is NP-complete to test whether a graph is a cover graph. Before 1940, G. Birkhoff posed the following open problems. (1) Compute for small n all non-isomorphic lattices/posets on a set of n elements. (2) Find asymptotic estimates and bounds for the rate of growth of the number of non-isomorphic lattices/posets with n elements. (3) Enumerate all finite lattices/posets which are uniquely determined (up to isomorphism) by their diagrams, considered purely as graphs. It is known that these problems are NP-complete. Recently, Brinkmann and McKay obtained the number of non-isomorphic posets and lattices with at most 18 elements. The work of enumerating all non isomorphic posets is still in progress. Thakare, Pawar and Waphare enumerated the non-isomorphic lattices containing n elements and up to n+1 edges. The work included in the Thesis is a contribution towards partial solutions to the above mentioned open problems. |
| URI: | http://hdl.handle.net/123456789/178 |
| Appears in Collections: | Ph.D Thesis |
Files in This Item:
| File | Description | Size | Format | |
|---|---|---|---|---|
| Ashok Bavale Sir.pdf | Ashok Bavale Sir's Ph.D Thesis | 1.37 MB | Adobe PDF | View/Open |
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