On left permutable inverse LA-semigroups

We investigate some fundamental properties of left permutable inverse LA-semigroups in which inverses commute. We prescribe a natural partial order and order ideal on an inverse LA-semigroup. Then we use the partial order to show that the set of idempotent elements in a left permutable inverse LA-semigroup is an order ideal. Also, we introduce the compatibility relations to ascertain important characteristics of meets and joins in the left permutable inverse LA-semigroups. Finally, we establish the conditions under which a left permutable inverse LA-semigroup is infinitely distributive.