2 Sentences With "quasiorders"

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A biordered set ("boset") is a mathematical object that occurs in the description of the structure of the set of idempotents in a semigroup. The concept and the terminology were developed by K S S Nambooripad in the early 1970s. The defining properties of a biordered set are expressed in terms of two quasiorders defined on the set and hence the name biordered set. Patrick Jordan, while a master's student at University of Sydney, introduced in 2002 the term boset as an abbreviation of biordered set.
Well-founded induction can be used on any set with a well-founded relation, thus one is interested in when a quasi-order is well-founded. (Here, by abuse of terminology, a quasiorder \le is said to be well-founded if the corresponding strict order x\le y\land y leq x is a well-founded relation.) However the class of well-founded quasiorders is not closed under certain operations—that is, when a quasi-order is used to obtain a new quasi-order on a set of structures derived from our original set, this quasiorder is found to be not well-founded. By placing stronger restrictions on the original well- founded quasiordering one can hope to ensure that our derived quasiorderings are still well-founded. An example of this is the power set operation.