Lattice properties in dioids

Abstract

A dioid (R, +, ·) means an idempotent semiring with additive identity ‘ε’ and multiplicative identity ‘e’. In dioid if the multiplicative identity ‘e’ is an absorbing with respect to +, i.e., a + e = e + a = e, then addition distributes over multiplication. In this case the dioid is complete lattice and distributive. Using these we prove that a complete dioid is always a distributive lattice.