#### Name

Titov Georgy Nikolaevich

#### Scholastic degree

•

#### Academic rank

associated professor

#### Honorary rank

â€”

#### Organization, job position

Kuban State University

#### Web site url

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## Articles count: 2

There is a widely known problem regarding the
ordering of the partially ordered sets (Linear Ordering
Problem). It boils down to finding the numerations of
such sets. The main result of this article is a
generalization of one of the known S. S. Kislitsyn's
results about finding the number of numerations of
finite partially ordered sets

Partially ordered set is a basic concept of modern settheoretic
mathematics. The problem of linear set
ordering with given binary relations is well-known.
Every partial order over a finite set can be linearly
ordered, but not every binary relation over this set can
be linearly ordered as well. Up to now, there is no
known formula for calculating the number of partial
orders over a given finite set. It appears that there is a
formula for calculating linearly ordered binary
relations over a finite set. This article is concerned
with derivation of this formula. The fact from work of
G.N. Titov [9] that a binary relation over a finite set is
linearly ordered if and only if any diagonal block,
derived from the binary relation matrix as a result of
setting main diagonal elements to zero, contains at
least one zero row (diagonal block of matrix means
any matrix composed of elements at the crossings of
rows and columns of a given matrix with the same
numbers), plays a key role in process of corroboration.
The main conclusion of the article is a theorem that
allows to find the number of linearly ordered binary
relations over a set of n elements using the formula. A
recurrence formula for the number of linearly ordered
(irreflexive) binary relations over a finite set of n
elements, provided in the lemma, was derived as well