The order relation [less than or equal to] in quotient set [OMEGA]/ ~ is elicited by the incomplete preference relation [less than or equal to] in E.

The order relation [less than or equal to] in quotient set [OMEGA]/ ~ which is elicited by the incomplete preference relation [less than or equal to] in E is a partial order.

Assuming that [OMEGA]/ ~ = {[x], x [member of] [OMEGA]} is a quotient set corresponding to the equivalence relation ~, then applying Lemmas 8 and 9, we get that the order relation [less than or equal to] in quotient set [OMEGA]/ ~ which is elicited by the incomplete preference relation [less than or equal to] in [OMEGA] is a partial order.

If the union of two fundamental open sets is a fundamental open set the

quotient set (H/R m , ) where O(x) O(y) = O(x) O(y) is a semigroup.

The set c/~ of all equivalence classes [x] of c is called the quotient set derived from c.

p is the projection of the magnitude set c onto its quotient set c/~.

Suppose (c, [less than or equal to], [circle]) is the commodity relational structure satisfying axioms A(1.1) - A(1.5) and suppose c/~ is the quotient set factored from c, using the equivalence relation ~.

Thus, the measurement properties that originate in the magnitude set c are preserved both in [r.sup.*] and in the quotient set c / ~ in such a way that the numeric sturucture in [r.sup.*] and the empirical structure in the quotient set c/~ are measurement copies of each other.

Then we compute the cardinality of the different quotient sets. Finally, the algebraic aspect is tackled: we first study sets defining quotient algebras of FQSym, and then show that these algebras are free.

Given the four types of statistics on [G.sub.n], we construct some quotient sets on [G.sub.n] as follows: consider a partition ([A.sub.1], ..., [A.sub.p]) of {P, V, Dr, Dd}.

3.4 The quotient sets (P [union] V [union] Dd, Dr) and (P [union] V [union] Dr, Dd)

3.5 The quotient sets (P, Dr [union] V [union] Dd) and (V, Dr [union] P [union] Dd)