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By Birger Iversen

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1) Every natural number is cancellable under addition; every natural number ^ 0 is cancellable under multiplication. (2) In a lattice there can be no cancellable element under the law sup other than the identity element (least element) if it exists; similarly for inf. In particular, in the set of subsets of a set E, 0 is the only cancellable element under the law U and E the only cancellable element under the law D. Proposition 2. The set of cancellable (resp. left. Cancellable, resp. right cancellable) elements of an associative magma is a submagma.

V v e H. Clearly (a) implies (d). v1 e H (no. 1 ). Remarks. (1) Similarly it can be shown that condition (b) is equivalent to the condition (C) II * 0 and the relations x e II and y e H imply y~1x e H. H = H and H1 = H. H <= HandH'1 <= Hby (b). H = Handtaking inverses transforms the inclusion H-1 <= II into H c H"1, whence formulae (1). If H is a stable subgroup of G and K is a stable subgroup of H, clearly K is a stable subgroup of G. The set jej is the smallest stable subgroup of G. The intersection of a family of stable subgroups of G is a stable subgroup.

Thenf is compatible with R. Let T be the equivalence relation associated with f. Then ua = y, (modT) for all a e 1 and T is compatible with the law on M, hence T is coarser than R; this proves the proposition. § 2. IDENTITY ELEMENT; CANCELLABLE ELEMENTS; INVERTIBLE ELEMENTS 1. IDENTITY ELEMENT Definition 1. Under a law cf composition T on a set E an element e of E is called cm identity element iff or all x e I i, e T x = x T e = x. There exists at most one identity element under a given law T, for if e and e' are identity elements then e = e T e' = e'.