Unitization of a lattice ordered ring with a truncation

Abstract

Let R be a lattice ordered ring along with a truncation in the sense of Ball. We give a necessary and sufficient condition on R for its unitization RQ to be again a lattice ordered ring. Also, we shall see that RQ is a lattice ordered ring for at most one truncation. Particular attention will be paid to the Archimedean case. More precisely, we shall identify the unique truncation on an Archimedean ℓ-ring R which makes R Q into a lattice ordered ring.

Key words: Alexandroff unitization, Archimedean, lattice ordered group, lattice ordered ring, orthomorphism, Stone f-ring, truncation, truncation unit, lattice homomorphism, unitization, weak unit.