Direct products of cyclic semigroups with zero, admitting outerplanar and generalized outerplanar Cayley graphs
Д. В. Соломатин · Vestnik Tomskogo gosudarstvennogo universiteta Matematika i mekhanika · 2024
The article presents the characteristic properties of direct products of semigroups with zero admitting outerplanar Cayley graphs, as well as their generalizations in the defining relations of copresentation. Theorem 1. A finite semigroup S with zero that is a direct product of nontrivial cyclic semigroups with zero admits an outerplanar Cayley graph if and only if one of the following conditions holds: 1) 0 0 3 2 1 h h S a a a b b b + = = where h is a natural number and h < 4; 2) 1 2 1 2 0 0 0 1 n r r i i i i S a a a a a a + + = = = where r and n are natural numbers and r 2 ; or r = 3, n = 1; 3) 0 0 r m r 2 S a a a b b b + + + = = where r and m are natural numbers and m 2 ; 4) 0 1 2 0 0 0 1 n r r i i i i S a a a a a a + + = = = where n = 1; or r = 1, n = 2. Theorem 2. A finite semigroup S with zero that is a direct product of nontrivial cyclic semigroups with zero admits a generalized outerplanar Cayley graph if and only if one of the following conditions holds: 1) 0 0 r m r h t h S a a a b b b + + = = where for natural numbers r, m, h, t one of the following restrictions is satisfied: 1.1) r = 2, m = 1, h < 4, t = 1; 1.2) r = 3, m = 1, h = 3, t = 1; 2) 1 2 1 2 0 0 0 1 n r r i i i i S a a a a a a + + = = = where r and n are natural numbers and r 3 ; 3.1) 0 2 1 2 2 1 2 S a a a b b b + + + = = ; 3.2) 0 0 r m r 2 S a a a b b b + + + = = where r and m are natural numbers and m 2 ; 4) 0 1 2 0 0 0 1 n r r i i i i S a a a a a a + + = = = where n = 1; or r = 1, n = 2.