Finite-dimensional Archimedean lattice-ordered algebras with certain d-elements
by
Jingjing Ma
University of Houston at Clear Lake

Two typical examples of finite-dimensional l-algebras over a totally ordered field mentioned by G. Birkhoff and R.S. Pierce in their 1956’s paper are finite group l-algebras with the coordinate-wise order and matrix l-algebras with the usual order. We characterize those finite-dimensional l-algebras that are isomorphic to a finite direct sum of group l-algebras and matrix l-algebras. An important class of l-rings introduced by G. Birkhoff and R.S. Pierce in the same paper is the class of regular l-rings. An l-ring R is called regular if the regular representation of R is an l-representation. We characterize some finite-dimensional real l-algebras which are also regular. A positive element e in an l-ring R is called a d-element if the multiplications by e are l-endomorphisms of the underlying l-group R. In above characterizations, certain d-elements play an important rule.

Date received: January 31, 2002