Symmetric Complement Research Articles

Lexique-Grammaire des Adjectifs Symétriques

In this paper, we examine the adjectives that seem to require two arguments in symmetric relation. To syntactically define these elements, we observed whether an adjective can appear in the following transformational relation or not: N0 N1-wa Adj = N1 N0-wa Adj. Those that allow this relation are classified in the class AWS and named Symmetric Adjectives. They constitute a semantically and syntactically homogeneous class. Two types of problematical structutures were considered: one is the construction containing a complement in -ei, replaceable by a symmetric complement in -wa; the other is the construction with complement in -wa which is not a symmetric one. Besides, we came across one more complicated structure concerning symmetric construction, that is, N0 N1-wa N2-i Adj. However, it cannot be considered as the basic structure for symmetric adjectives, since N2-i is either a restructured element or a symmetric noun: in the first case, this structure is a derived one; in the second case, it is this noun that requires a symmetric complement Nj-wa, and not the adjective. Therefore, this structure cannot characterize symmetric adjectives. In this way, we obtained 120 adjectives for the class AWS among 5300 Korean simple adjectives. The result of our study is represented in the form of a matrix in the Appendix. More detailed studies on symmetric verbs and nouns are also required as well as more advanced analyses about semantic and logic constraints in the distribution of arguments.

Unique nontransitive additive conjoint measurement on finite sets

Nontransitive additive conjoint measurement for a binary relation>on a setX1×X2×...×X n ofn-tuples(x1,...,x n ), (y1,...,y n ),... is concerned with the representation $$(x_1 ,...,x_n ) > (y_1 ,...,y_n ) \Leftrightarrow \sum\limits_{i = 1}^n {\phi _i (x_i ,y_i ) > 0,} $$ whereφ φ is a skew symmetric real valued function onX i ×X i . The representation is said to be unique if, whenever (φ1,...φ n ) and (φ 1 ′ ,...φ n ′ ) satisfy it, there is a positive constant λ such thatφ i ′ =λφ i for alli. This paper investigates unique representation for nontransitive additive conjoint measurement when everyX i is finite. It begins with background on measurement theory, followed by conditions for uniqueness that are based on equations ∑φ i (x i ,y i )=0 that correspond to the symmetric complement of >. We then examine aspects of sets of unique solutions forn=2, for arbitraryn with |X i |=2 for alli, and forn=3. Various combinatorial and number-theoretic arguments are used to derive results that count and characterize sets of unique solutions.

Operations on binary relations

The effects of five basic operations (asymmetrization, complementation, dualization, symmetrization, transitive closure) on binary relations are examined. Identifies between compound operations are developed (e.g. the symmetric part of the transitive closure of the complement of the transitive closure equals the transitive closure of the symmetric complement of the transitive closure), ordering aspects of compound operations are noted, and it is shown that in addition to the empty and universal relations at most 110 different relations can be generated from a given binary relation by sequential applications of the five basic operations. Moreover, 110 is the least upper bound, and none of these 110 requires more than seven applications of the basic operations for its expression. One of the potentially irreducible compound operations of length seven is cstcatc, the complement of the symmetric part of the transitive closure of the complement of the asymmetric part of the transitive closure of the complement.