THE SEMIGROUP OF NONEMPTY FINITE SUBSETS OF INTEGERS

Let Z be the additive group of integers and g the semigroup consisting of all nonempty finite subsets of Z with respect to the operation defined by A B {a+b a e A, b E B}, A, B e . For X , define X to be the basis of <X-min(X)> and X the basis of <max(X)-X>. In the greatest semilattice decomposition of g, let (X) denote the archimedean component containing X and define Go(X) [Y (X) min(Y) 0}. In this paper we examine the structure of and determine its greatest semilattice decomposition. In particular, we show that for X, Y (X) (Y) if and only if AX Ay and X By. Furthermore, if X g is a non-singleton, then the idempotent-free (X) is isomorphic to the direct product of the (idempotent-free) power joined subsemigroup Go(X) and the group Z.

The semigroup is clearly commutative and is a subsemigroup of the power semigroup of the group of integers, (the semigroup of all nonempty subsets of Z). In this paper we will determine the greatest semilattice decomposition of and describe the structure of the archimedean components in this decomposition. As we will soon see, there is a surprisingly simple necessary and sufficient condition for two elements to be in the same component. In the greatest semilattice decomposition of 8, let (A) denote the archimedean component containing A. As usual, define the partial order < on the (lower) semilattice as: (A) < (B) if and only if nA B C for some C g 8 and n e Z+ (equivalently: X Y e (A) for some (all) X (A) and Y e (B)).
We refer the reader to Clifford and Preston [2] and Petrich [3] for more on the greatest semilattice decomposition of a commutative semigroup. Observe that since 0 is the only idempotent and indeed the identity, (A) is idempotent-free if A is a non-singleton, ((0) consists of all the singletons in 8 and in fact (0) Z).
Furthermore, if follows that the subgroups of 8 are of the form {{gx} x e Z}, where g is a non-negative integer. Finally, note that 8 is clearly countable, but this of course does not imply that there are also infinitely many archimedean components. However, as will soon be shown, there are in fact infinitely many components.

GREATEST SEMILATTICE DECOMPOSITION.
For X e , define A X to be the basis of <X-min(X)> and B X the basis of <max(X)-X>. Note that A X B X {0} if and only if X is a singleton. Also observe that A X is a finite set with at most a elements, where a is the least positive integer in A X (if A X {0}), and similarly for B X. Since gcd(X-min(X)) gcd(X-max(X)), it follows that in general gcd(A X) gcd(Bx).
Given sets A and B, it is clearly not always possible to find an X such that A X A and B X B. However, we do have a positive result. First we need the following lemma. LEMMA 2.1. Let S be a positive integer semigroup with respect to addition.
The following are equivalent. (i) S contains m such that x > m implies x S.     [c-a+1, ma It is clear that if u e qU with u < c-a and q > n, then a-1 u e u {x e A.
x < c-a}.  G(X); that is, G(X) < (Y) and the proof is complete.
Observe that clearly Ay A X and By B X is a sufficient condition for (I(X) G(Y). However, it is not a necessary condition (see Spake [4]). Since Ay and By are finite sets, it is relatively easy to determine when G(X) G(Y) via Theorem 2.4 (ii). Also, as the trivial case of Theorem 2.4, we have G(0) for all X e and hence (0,I) is an ideal of Define Go(X) {Y e G(X) min(Y) 0} and note that Go(X) is a subsemigroup of G(X). Moreover, since elements of G(X) can be uniquely expressed in the form U a, where U e Go(X) and a e Z, evidently G(X) O(X) Z. Recalling the proof of Theorem 2.3, apparently if X is a non-singleton, then G0(X) is power joined. We therefore immediately have THEOREM 2.5. The idempotent-free archlmedean componen.t. O.(X), where X non____-singleton, is isomorphic t__o th___e direc____t product of th__e Idempotent-free power joined subs..emigroup (Io(X) and the group Z.

STRUCTURE OF THE COMPONENTS.
The structure of (0) is clear, since (0) Z. In this section we investigate the structure of (X) when X is a non-singleton. We begin with a general result from Theorem 2.3. Next we reproduce several definitlons and facts from Tamura [5] that we will need in the following development. We direct the reader to [5] for a more complete discussion of the notions which follow. Let T be an additively denoted commutative idempotent-free archimedean semigroup. Define a congruence Pb on T, for fixed b, as x Pb y if and only if nb x mb y for some n, m g Z+.
Then T/b G b is a group called the structure group of T determined by the standard element b. Also, define a compatible partial order < on T as follows: x < y if and only if x =nb y for some n e Z+. semilattice with respect to <. In fact, for each G b, T forms a discrete b tree without smallest element with respect to <, (a discrete tree, with respect to b <, is a lower semilattice such that for any c < d the set{x c < x < d} is a finite chain). Finally, we define a relation n on T as follows: x y if and only if nb+ x =nb + y for some n Z+.
The relation is the smallest cancellatlve congruence on T.
We continue our development with the following theorem. (mod m)} is a discrete tree without smallest element with respect to <.  Conversely, if U rA V + sA for some r, s E Z+, then max(U) + rgm--max(V) + sgm.
Since glmax(U) and glmax(V), evidently max(U)/g m max(V)/g (mod m). By Proposition 3.1, if t max {max(A C) + d-b+1, max(B C) c-a+1} and t E Z+, then X A C u (t B C) e G0(C) with max(X) to It follows that for each e Zm, there exists X e @0(A) with max(X)/g m (mod m). Therefore, the structure group of o(A) determined by the standard element A is Zm.
Using the above, it is clear that for X, Y e (A), X rA Y sA for some, r, s e Z+ if and only if min(X) min(Y) and (max(X)-min(X))/g -= (max(Y)-min(Y))/g (mod m).
This completes the proof.
We conclude this paper with two related propositions. PROOF. First, observe that o(X) and (Io(U) have isomorphic greatest cancellative homomorphic images, since (,o(X) (Io(U). Let V e o0(U) and max(V). Since A U c_ V and B U c_ t V, it follows that t > max {max(Au), max(Bu)}. Moreover, using Proposition 3.1, t x e <Bu> and t y e <Au> for all AU, y e B U. Thus, by the definition of c. and d., t e C. Furthermore, if C then evidently A U (r-B U) e (o(U). Consequently, the proof is complete by