Positively ordered 0-skewsemifields

Abstract

A triple (K, +, ํ) is called a 0-skewsemifield if 1) (K, ํ) is group with zero 0, (K, +) is a semigroup, 3) for all x, y, zEK, x(y+z) = xy+xz and (y+z)x = yx+zx, and 4) for every xEK, x+0 = 0 = 0+x. For a 0-skewsemifield K, let K* denote K/{0}. A quardruple (K, +, ํ, <) is called a positively ordered 0-skewsemifield if and only if (K, +, ํ) is a 0-skewsemifield and < is a partial order on K such that for all x, y, zEK 1) x0. The subset P = {xEK/x>1} of a positively ordered 0-skewsemifield is called the positive cone of K. Let {Ki/iE/} be a family of 0-skewsemifields. The direct product of the famil {Ki/iE/} is the set of all elements (xi)iEi in the cartesian product of family {Ki*/iE/} and 0 where 0 = (0i)iEi together with the componentwise operations. Let L be a subskewsemifield of the direct product of {Ki/iE/}. L is said to be a subdirect product of {Ki/iE/} if and only if for every jE/, IIj(L) = Kj where IIj is the natural projection map. A positive lattice 0-skewsemifield K is said to completely integrally closed if and only if for every aEK, if there exists a bEK such that an