ENUMERATION OF CERTAIN ALGEBRAIC SYSTEMS AND RELATED RESULTS

Bhavale, Ashok N.; Waphare, B.N. (Guide)

dc.contributor.author	Bhavale, Ashok N.
dc.contributor.author	Waphare, B.N. (Guide)
dc.date.accessioned	2020-02-13T05:02:54Z
dc.date.available	2020-02-13T05:02:54Z
dc.date.issued	2013-12
dc.identifier.uri	http://hdl.handle.net/123456789/178
dc.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.	en_US
dc.language.iso	en	en_US
dc.publisher	UNIVERSITY OF PUNE	en_US
dc.subject	Mathematics	en_US
dc.subject	ALGEBRAIC SYSTEMS	en_US
dc.subject	Birkhoﬀ’s open problems	en_US
dc.subject	Structure theorem	en_US
dc.subject	Nullity of a poset	en_US
dc.subject	Ore’s open problem	en_US
dc.subject	Whitney type characterization	en_US
dc.subject	Enumeration of lattices	en_US
dc.subject	Basic blocks	en_US
dc.subject	Counting fundamental basic blocks	en_US
dc.subject	Enumeration of blocks on six reducible elements	en_US
dc.subject	Enumeration of blocks on ﬁve reducible elements	en_US
dc.subject	Enumeration of blocks on four reducible elements	en_US
dc.title	ENUMERATION OF CERTAIN ALGEBRAIC SYSTEMS AND RELATED RESULTS	en_US
dc.type	Ph.D Thesis	en_US