Containment for c, s, t operators on a binary relation |
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Authors: | Frank D Farmer |
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Institution: | (1) Department of Mathematics, Arizona State University, 85287 Tempe, AZ, USA |
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Abstract: | The operators c, s and t are complement, symmetric and transitive closure
of a binary relation. If u and v denote finite sequences
of these operators then we define
u v iff for every binary relation
. We find the distinct representative and
containment between these sequences. The asymmetric operator is not one of these. There
are 54 representatives for binary relations, 20 for transitive relations, and 10 for symmetric
relations. There are 26 component types of a binary relation, 10 for transitive relations, and
6 for symmetric relations. There are 16 connected types of a binary relation, 8 for transitive
relations, and 4 for symmetric relations. We study well founded relations. Total relations
may not be contractible but well founded ones are. The complement of (a Hasse diagram
of) a non-empty partial order of arbitrary cardinality is contractible. Ordered sets are
naturally homotopy equivalent to partially ordered sets. There are 10 relations which can
have arbitrary polyhedral homotopy type and 42 are either contractible or the homotopy
type of a wedge of n-spheres. The homotopy type of two relations is not determined. |
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Keywords: | 06A06 05C20 05E25 18B10 57M15 03E02 05C10 55P10 06A07 |
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