Algebraic Structures



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Algebraic Structures

(defrelation semigroup (?s ?f ?o) := (and (binop ?f ?s) (associative ?f ?s) (identity ?o ?f ?s)))

(defrelation abelian-semigroup (?s ?f ?o) := (and (semigroup ?s ?f ?o) (commutative ?f ?s)))

(defrelation group (?s ?f ?o) := (and (binop ?f ?s) (associative ?f ?s) (identity ?o ?f ?s) (invertible ?f ?o ?s)))

(defrelation abelian-group (?s ?f ?o) := (and (group ?s ?f ?o) (commutative ?f ?s)))

(defrelation ring (?s ?f ?o ?g ?i) := (and (abelian-group ?s ?f ?o) (semigroup ?s ?g ?i) (distributes ?g ?f ?s)))

(defrelation commutative-ring (?s ?f ?o ?g ?i) := (and (abelian-group ?s ?f ?o) (abelian-semigroup ?s ?g ?i) (distributes ?g ?f ?s)))

(defrelation integral-domain (?s ?f ?o ?g ?i) := (and (commutative-ring ?s ?f ?o ?g ?i) (operation ?g (difference ?s (setof ?o)))))

(defrelation division-ring (?s ?f ?o ?g ?i) := (and (ring ?s ?f ?o ?g ?i) (binop ?g (difference ?s (setof ?o))) (invertible ?g (difference ?s (setof ?o)))))

(defrelation field (?s ?f ?o ?g ?i) := (and (division-ring ?s ?f ?o ?g ?i) (commutative ?f ?s)))

(defrelation partial-order (?s ?r) := (and (irreflexive ?r ?s) (asymmetric ?r ?s) (transitive ?r ?s)))

(defrelation linear-order (?s ?r) := (and (irreflexive ?r ?s) (asymmetric ?r ?s) (transitive ?r ?s) (trichotomizes ?r ?s)))



Vishal I. Sikka
Wed Dec 7 13:23:42 PST 1994