Partition Relations and Transitivity Domains of Binary Relations (original) (raw)
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First of all, I would like to thank my advisor, Prof. Ottavio D'Antona, who encouraged me to go on with my studies. To him and to Dr. Vincenzo Marra my sincerest thanks for their patient guide, their indispensable teachings, their constant supporting, and the fruitful exchange of ideas and source of inspiration. I also would like to thank the Ph.D. coordinator, Prof. Giovanni Naldi, and Prof. Lluis Godo, my tutor during my first year of course in Barcelona. My gratitude to my family who has always supported me in these years, and all the friends who were close to me in the last months when I was writing this thesis.
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In the present paper, we introduce and study the continuity and some properties for a set equipped with a transitive binary relation which we call t-set. Also, we give a characterization of a continuous directed complete posets via continuous t-sets. Furthermore, some properties of algebraic t-sets are considered. Our work is inspired by the slogan: "Order theory is the study of transitive relations" due to Abramsky and Jung [1]. The corresponding results of our results due to Nino-Salcedo [8], Heckmann [4], and Zhang [11] are generalized.
There are well-known isomorphisms between the complete lattice of all partitions of a given set A and the lattice of all equivalence relations on A.
Completeness properties of transitive binary relational sets
Cornell University - arXiv, 2020
The present paper is devoted to study some completeness properties of transitive binary relational set, i.e., a set together with a transitive binary relation (so called t-set). 1 Introduction Abramsky and Jung [1] introduced a method to construct a canonical partially ordered set from a pre-ordered set and said: "Many notions from the theory of order sets make sense even if reflexivity fails". Finally, they sum up these considerations with the slogan: "Order theory is the study of transitive relations". Heckmann [3] introduced and studied the concepts of bounded complete poset, bounded complete domain, finitely complete poset, complete domain, finitarily complete poset, strongly compactly complete domain and compactly complete domain. This paper is organized as follows: In Section 2, some definitions and results concerning some completeness properties of poset and domain were presented. In Section 3, bounded complete t-sets and bounded complete domain t-sets were introduced and studied. In Section 4, finitely complete t-sets and complete domain t-sets were introduced and studied. In Section 5, we extend the concept of finitary sets in transitive binary relational sets and then introduced and study the concept of finitarily complete t-sets. Finally, in Section 6, strongly compactly complete t-sets and compactly complete t-sets were introduced and studied.
Discrete Mathematics, 1993
Critically indccomposahic partially ordered sets. graphs, tournaments and other binary relational struc*ures. Discrete Mathematics 113 (1093) 191-205. A hnite. indecomposabic partially ordered set is said to be critically indecomposahlc if. whenever an clement is removed. the resulting induced partially ordered set is not indccomposabic. The same terminology can be applied to graphs. tournaments, or any other relational structure whose relations are binary and irrcflexive. It will be shown in this paper that critically indecomposable partially ordered sets are rather scarce; indeed. there are none of odd order. there is exactly one of order 4. and for each even k 3 6 there are exactly two of order k. The same applies to graphs. For tournaments, there are none have even order, there is exactly one of order 3, and for each odd k B 5 there are precisely three of order k. In general, for arbitrary irreflexivc binary relational structures, we will XC that ail critical indecomposabics fail into one of nine infinite classes. Four of these classes are even-they contain no structures of odd order and for cvcn k 2 6 they each contain (up to a certain type of equivalence) exactly one structure of order k. The five other classes are odd-they contain no structures of even order and for each odd k 35 they each contain cxactiy one structure of order k. From this characterization of critically indccomposabic strucrures. it will bc evident that ail indccomposabfc substructures of critically indccompos;rbic structures are themselves critically indccomp()sabfc. Finally, it is proved that every indecomposable structure of order 11 + 2 (tt 3 5) has an rndccomposabic substructure of order tt