Jónsson posets and unary Jónsson algebras

摘要

We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than ({|P|}) and ({kappa}) is a cardinal number strictly less than ({|P|}), then P has a principal order ideal of cardinality at least ({kappa}). We apply this result to characterize the possible sizes of unary Jónsson algebras.