The inclusion-exclusion principle without distributivity

Inspired by the article of P. Grzegorzewski [The inclusion-exclusion principle for IF-events, Inform. Sci. 181 (2011), 536-546], who has worked two generalizations of the inclusion-exclusion principle for IF-events, a generalization of the inclusion-exclusion principle for mappings with values in semigroups is presented here. The main idea is in replacing the distributivity and idempotency laws, by one new axiom.