NEW COMPOSITE RELATIONS CHARACTERIZATION OF STAR-LIKE FINITE SEMIGROUPS

NEW COMPOSITE RELATIONS OF THE STAR-LIKE SEMIGROUP

Abstract

The study establishes new star-like classical finite semigroups, provides a cohesive explanation, appraises some combinatorial composite relations on $\alpha\omega_n^*$, and proves some combinatorial relations of star-like $P\omega_n^*$, $T\omega_n^*$, and $O^{cd}P\omega_n^*$ star-like-partial ordered connected transformation semigroups. Let $\alpha\omega_n^*$ be a star-like transformation semigroup with a star-like diskpoint $m^*$, diagonal difference operator $\nabla_{b_{(n, m^*)}}^*$, and vertical difference operator $\Delta_{(n, r^*)}^*$. The study shows that star-like sequences converge uniformly and are reducible if $|\nu^*(\alpha_{(i,j)}^*)| = F(n, \nu^*)$. Also, for a star-like diskpoint $m^*, $ $F(n, m^*) = (n-1)^{2} + (m+1)(n-2)$ and $|\Delta_{(n, r_3^*)}^*| = \lambda_{i}^{2*} + (n - 2) $ such that $|c^{+}(\alpha^*)| \leq |c^{-}(\alpha^*)|$. The new combinatorial composite relations were generated from the existing ones by composition of star-like mapping, which was applied to star-like $\alpha\omega_n^*$ finite semigroups with emphasis on combinatorial triangular arrays.