On properties of posets of MM-type (1, 2, 7)

Abstract

We calculate the coefficients of transitiveness for all posets of MM-type (1, 2, 7) (i.e. posets, which are minimax equivalent to the poset (1, 2, 7)).

Класифікація Частково Впорядкованих Множин, MM-тип Яких Дорівнює Симетричній Надсуперкритичній Множині Порядку 9

“…We put 𝑛 𝑤 = 𝑛 𝑤 (𝑆) := |𝑆 2 < | and denote by 𝑛 𝑒 = 𝑛 𝑒 (𝑆) the number of pairs of neighboring elements. The ratio 𝑘 𝑡 = 𝑘 𝑡 (𝑆) of the numbers 𝑛 𝑤 − 𝑛 𝑒 and 𝑛 𝑤 are called the coefficient of transitiveness of 𝑆; if 𝑛 𝑤 = 0 (then 𝑛 𝑒 = 0), we assume 𝑘 𝑡 = 0 (see [20]).…”

“…Let 𝑃 be a x poset. A poset 𝑆 is called of 𝑀 𝑀 -type 𝑃 if 𝑆 is minimax isomorphic to 𝑃 [20]. In the case when the poset 𝑃 is an oversupercritical one we say that 𝑆 is of oversupercritical 𝑀 𝑀 -type.…”

Combinatorial properties of non-serial posets with positive Tits quadratic form

“…This paper is devoted to study of combinatorial properties of posets with positive Tits quadratic forms (which clarify and generalize some calculations in a partial case [10]).…”