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By Charles Benedict Thomas

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Now suppose that aj at < b for some b e S and for all i G /. Then di < a,ib < ab for all i E I. Thus a = V a,i < ab. But then a = (aa~1)ab — ab, so that a~xa = a~lab. Hence a~la < b. It follows that V a~lai = a~1a. ■ We now consider how meets and joins behave with respect to multiplication. For joins, the strongest result we can prove is the following. 28 Introduction to inverse semigroups Proposition 18 Let S be an inverse semigroup, A = {a^: i £ 1} a non-empty subset of S and s £ S. (1) Ifa = \/ai andaia^1 < s~xs for alii £ / then V sai exists and sa = V sa i- (2) If a = V a i anda^lo,i < ss~l for alii £ / then \J a{S exists and as = \ / a i s Proof We prove (1); the proof of (2) is similar.

Then < is said to be compatible with the multiplication if for all a,b,c,d € S we have that a < b and c < d implies that ac < bd. In this case, the semigroup S is said to be partially ordered by <. Observe that if T is an inverse subsemigroup of S, then the natural partial order defined in T agrees with the restriction to T of the natural partial order on S. Let (P, <) be a poset. If z < x,y then z is said to be a lower bound of x and y. If z is the largest of the lower bounds it is called the greatest lower bound and denoted by x A y.

I am grateful to Ian Stewart for supplying the references to Mackay's work. Some attempts at alternative algebraic descriptions within crystallography were also carried out by Fichtner in relation to polytypic structures (layered structures, such as silicon carbide) [77], [78], [79] and [80]. I am unable to judge the significance of Fichtner's work, but it contains explicit references to partial symmetries in the context of crystallography. Weinstein's article [431] is the most recent one in which the identification of group theory and symmetry is questioned.

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