ON LATTICE-DILATIONS AND CONTRACTIONS IN f-RINGS

ABSTPACT . This paper splits broadly into two related parts, concerned respectively with generalized idempotents(associated to superunities and subunities) and with lattice-dilations and contractions in an f-ring A. If u is a superuuiity, we characterize the mappings F:A-->A satifying IF(x)-F(y)I = uIx-y1 (u-dilations) as the mappings of the form F(x)= x(u-2e)tb, with e being a generali .zed idempotent, and obtain an analogous result for lattice-contractions . The set of the hcnageneous ones(both cases) are proved to be Boolean algebras .

1 .Generalized ideaqotents .In an -¿-ring A, we introduce the following definitions : a) if u is a su T neruni ty, then e E A is a u-idempotent if eu = ue = e2 .b) if s is a subunity, then e E A is an s-idempotent if es = se = e2 .The respective sets will be denoted by I(u) and I(s) .
We first show that in an f-ring A , I (p) = { x E A 1 x A (p-x) = 0 } if p is a subunity or a superunity, and obtain as a consequence : Theorem 1 .I(p) is a Boolean algebra with the ordering of the ring .We pay considerable attention to projectable f-rings, that is, those for which al ® -L-= A holds for every a E A. They are specially interisting in this context in view of the following result for f-rings with a superunity u: Theorem 2 .A is projectable if and only if the polar of every element is the polar of a unique u-idempotent .
For the "only if" part of Th .2, it suffices proving that if aE A, then al = el , being e the projection of u onto á" «L .
Some ccepleteness and projection properties :L-,ply that the f-ring A be projectable .For instance, those of the main inclusion theorem [4J , and others .We have proved here that if I(u) is eonvex, then A is projectable .Frcm Th.2 we obtain that if A is projectable and non totally ordered, then I(u) is non trivial .
Sane results suggest the interest of studying the subset Pu (A) = t e-'-1 e E I(u)} of all the polars P(A) of the ring.In this connexion we prove: Theorem 3 .With the ordering of P(A), Pu (A) is a Boolean subalgebra of P(A), iscmorphic to the algebra I(u) .Moreover, Pu(A) is a sub-lattice of PP(A), the lattice of all principal polars, and a sualgebra of the Boolean algebra of direct sum:nands .If P(A) is superatonic,then I(u) is finite .
If u' is another superunity, then I(u) and I(u') are isomrphic in the following cases : a) A is Dedekind-cadete ; b) I(u) and I(u') are complete Boolean algebras and are convex in A.
The proof of TM uses mainly the decomposition A = el ® (u-e)1 if eeI(u), and the fact that if el , e2 e I(u) and el = e2 then el= e2 .The isomr phism of the statement is given by I(u)--> Pu(A) , e~(u-e)1 .
By using Th .3 in the projectable case, we obtain : Theorem 4 .Let A be a projectable f-ring .a) I(u) is atomic in the following cases : 1) P(A) is atcenic ; 2) A is basic or ~letely distributive .b) If A has a finite basis, then I(u) is finite .

. Boolean al ras of lattice -dilations .
If u is a central supeninity of the f-ring A, we generalize the notion of ¿-isonetry ( [21 , [5'1, [6 1) by considering the mappings F : A -> A that satisfy IF(x)-F(y)j = uIx-y1 for every x,yE A. They will be called lattice-dilations or u-dilations , since IF(x)-F(y)I> 1x-y1 .We denote by Hu(A) the 'set of all hcmogeneous u-dilations, that is, those for which F(0) = 0 .On the other hand, if e E I(u), we consider the mapping cre:A -~A , ~e (x)= x(u-2e) and set Zu (A) _ {v'e leeI(u)1 The fundamental result we have obtained ncw follows : Theorem 7 enables us to give sane geometric interpretation of harogeneous u-dilaticns, especially by neans of the coneept of lattiee axial sycmtry .
Recall frcan [ 6 ] that if a e A , then f :A-s A. is a lattice axial symmetry of axis a if : 1) f is a group hamamrphism : 2) A=<aj® aL and 3) f La> I , f 1 a1= -I , with I being the identity mapping .Then we can partially rephrase TM : Every hcuegeneous u-dilation é is a homotecy of ratio u on orthogonal directions, followed by a lattice axial symmetry with respect to one of that directions(of axis u-e), or, which is the same : it is a lattice axial symmetry followed by a hanotecy of ratio u .Conversely, every lattice axial symmety, followed by a harotecy of ratio u is a hanogeneous u-dilation .
In the course of our study the set B of all square roots of u2 has naturally arisen .We have pMved that B =La¡ la ¡= u} .Moreover, Theorem 8 .With the same ordering of the ring, B is a Boolean algebra, that is isamorphic to the Boolean algebra I(u) .Hence isamorphic to Hu (A) .
The preceding iscmorphism is given by I(u) -B , e r-~2e-u .It is possible now to transfer to B many of the properties that could be asserted for the Boolean algebra I(u) .
2 .Boolean al ras of generalized idempotents .On account of I (p) having no special property whi.ch an arbitrary Boolean algebra need not nave, we have analized its boolean properties in eonnection with lattice and algebraic-theoretic properties of the ring .Since I (p) is closed by taking arbitrary suprema and infima, it is easy to relate the order oompleteness properties of A with ~lete ness properties of I(p) .
Theorem 5 .a) Hu (A)= o~)u (A) .b) Every u-dilation F is of the form F(x)= x(u-2e)T b, being b e A and e E I(u) .c) Every Fe Hu (A) is a hamtecy of ratio a, with I a I = u Part a)'of Th .5 has been proved by means of a suitable representation of A as a subdirect product of totally ordered rings, and the fact that for'a totally ordered ring with a superunity u, the only u-dilations are O'6 (x)= ux for every x, and a-u (x)= -ux for every x .Part c) follows on account of e E I(u) being a component of u . 1 Now, if we consider the Boolean ring structure of the Boolean algebra I(u),then Th .5 enables us to endow H u (A) with a Boolean structure : Theorem 6 .With the operations (ae ® é,) (x)= x(u-21e-e'I) and (oye ,Y ae' ) ( x) = x (u-2 (e n e' )), (Hu (A), , IK ) is a Boolean ring with unity, isomorphic to the Boolean ring I(u) .Now in view of the iscenorphism Hu(A)='I(u) and the theorems 3 and 4, we i have derived the corresponding properties of the Boolean algebra Hu (A), but we shall not explicitly mention them here .~ver, i.t is worth noting that if A is projectable and non totally ordered, then there exist non trivial u-iilations .The u-dilations v'o and o-u are interesting since we have : Theorem 7 .If F E Hu (A) , then there exists a unique decanposition A= B$C, with-B,C being ¿-ideals, for which FIB '7 0 and FI C Qu .Indeed, by Th .5 F= or e , for sane e e I(u), and it suffices taking B= e l and C= (u-e)l .
4 .Lattice-oontractive mappings .If s is a central subunity of the f-ring A, we can define "mutatis mutandis" the concept of latticecontraction (s-contraction) and hoimgeneous s-contraction by only interchanging u by s in the definition .With certain additional assumptions on A , it is possible to develop a theory for s-contractions, that is parallel to that of u-dilations, though less satisfactory in some aspects .The difficulty appears when serme of the properties that are valid for u-idemootents do not hold for s-idempotents .For instance, the decompositich A= el e (u-e)'L is no more valid for every e E I(s) .It remains valid however if A is Dedekind-complete or if s is a formal unity .Other properties still hold in absence of nonzero nilpotent elements .