On the linear orderability of two classes of finite semigroups (Q1188324)

The author proves two theorems: Theorem 1. Any (finitely generated) periodic semigroup \(S = \langle x_ 1, x_ 2, \dots, x_ s \rangle\) for \(s > 1\), with defining relations \[ (1) \quad x_ i = x_ jx_ i \;\text{ for } \;1 \leq i < j \leq s, \qquad (2) \quad x_ i^{m_ i+1} = x_ i^{m_ i} \;\text{ for } \;m_ i \in N \] is right linearly orderable but it is not linearly orderable. There exists a right order about which the semigroup \(S\) is left positively orderable. Theorem 2. Any finite semigroup \(S = \langle x_ 1, \dots, x_ s \rangle\) for \(s > 1\), with generating relations \[ (1) \quad x_ i = x_ j x_ i \;\text{ for } \;1 \leq i < j \leq s; \qquad (3) \quad x_ i^{m_ i} x_ j = x_ i^{m_ i} \;\text{ for } \;1 \leq i \leq j \leq s \;\text{ and } \;m_ i \in N, \] is positively linearly orderable.