Algebra univers. 41 (1999) 239–254 0002–5240/99/040239–16 $ 1.50+0.20/0 © Birkha¨user Verlag, Basel, 1999

Ideals of Priestley powers of semilattices J. D. FARLEY

Y Abstract. Let X be a poset and Y an ordered space; X Y (X Y S , X L ) denotes the poset of continuous order-preserving maps from Y to X with the discrete (respectively, Scott, Lawson) topology. If S is a –-semilattice, S s its ideal semilattice, and T a bounded distributive lattice with Priestley dual space P(T), it is shown that the following isomorphisms hold:

s

) (S P(T))s $ (S s)P(T) $ (S s)P(T . S L

Moreover, s

s)

) (S s)P(T $ (S s)P(T L

if and only if

s

s

) (S s)P(T =(S s)P(T ), L

and sufficient conditions and necessary conditions for the isomorphism to hold are obtained (both necessary and sufficient if S is a distributive –-semilattice).

1. Introduction Let S be a –-semilattice, M a bounded distributive lattice, and Pœ P(M) the Priestley dual space of M. The poset S P of continuous order-preserving maps from P to S with the discrete topology (ordered pointwise) is a Priestley power. In [3], Theorem 3.1, it is shown that if L is a lattice and P a finite poset, (L P)s $(L s)P,

Presented by Professor Ivan Rival. Received September 27, 1995; accepted in final form April 6, 1998. This work was supervised by Dr. H. A. Priestley, whom the author would like to thank. Dr. P. M. Neumann suggested helpful stylistic changes. This material is based upon work supported under a (U.S.) National Science Foundation Graduate Research Fellowship and a Marshall Aid Commemoration Commission Scholarship. 1991 Mathematics Subject Classification: 06B10, 06A12, 06B05, 06B35, 54A10, 54C40, 08A05, 06E15, 54F05. Key words and phrases: Function semilattice, ideal semilattice, semilattice, distributive lattice, Priestley duality, Scott topology, Lawson topology. 239

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where S s denotes the poset of ideals of a –-semilattice S (see also [9], Lemma 2). Jo´nsson and McKenzie have noted that the proof is valid for semilattices ([10], §5). Gra¨tzer and Schmidt have shown that, if L is a lattice and Con L its congruence lattice, Con(L P(M)) $ (Con L)P(Con M) if and only if either Con L is finite or M is finite. They have also shown that if E and A are complete distributive lattices such that E P(A) is complete, then either E satisfies the ascending chain condition or A has no infinite ‘‘complete-join independent antichain.’’ Moreover, if E is a distributive algebraic lattice, then E P(A) is complete (or algebraic) for every distributive algebraic lattice A if and only if E is finite. (See [9], Theorem 3, Theorem 5, and Corollary.) In §3, we show that (S P(M))s $(S s)P(M) , S the poset of continuous order-preserving maps from P(M) to S s where the latter has the Scott topology, and s

) (S P(M))s $(S s)P(M , L

where L denotes the Lawson topology (for the definitions refer to [7], Definitions II.1.3 and III.1.5). See Theorem 3.6 and Corollary 3.7. Note we are not assuming that S has a least element. We also find necessary conditions and sufficient conditions for the isomorphism (S P(M))s $(S s)P(M

s)

to hold, which are both necessary and sufficient if S is a distributive semilattice. See Proposition 3.26, Theorem 3.27, and Theorem 3.28.

2. Notation, definitions, and basic results For basic definitions and notation, consult [2]. If P is a poset and A ¤P, let A l be the set of lower bounds of A and A u the set of upper bounds. We denote the least element of a poset P (if it exists) by 0P or 0, and the greatest element (if it exists) by 1P or 1.

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Let 2 be the two-element chain. Let A and B be sets. If f: A “B is a function, let Im f denote its image f[A]. Let P be a poset. Let Q, R ¤P. Then ¡Q R œ{q  Q q 5r for some rR}. We also define ¡R œ ¡P R, ¡Q p œ ¡Q {p}, and ¡pœ ¡{p}, where pP. Dually we define  Q R,   R,  Q p, and  p, where pP. A down-set of P is a subset Q such that Q = ¡Q; dually we define an up-set. If P is an ordered space (by which we just mean a poset with a topology), O(P) is the lattice of open down-sets, U(P) the lattice of open up-sets, and D(P) the lattice of clopen up-sets, ordered by inclusion. A non-empty subset D of a poset P is directed if every finite non-empty subset of D has an upper bound in D. If a directed subset D has a supremum in a poset P, we denote it by  D. (The special notation, which is standard, serves as a convenient reminder that the set under consideration is directed.) An ideal is a directed down-set. The poset of ideals of P is denoted P s. An element k is compact in a poset P if, for every directed subset D whose supremum exists, k 5 D implies k5 d for some dD; the poset of compact elements is denoted k(P). An algebraic poset is a poset such that every directed set has a supremum and every element is a directed join of compact elements ([4], §1). An algebraic poset that is a complete lattice is an algebraic lattice; equivalently, P$ S s for a {0}-–-semilattice S. Algebraic posets such that every non-empty subset has a supremum are exactly the posets S s for a –-semilattice S; for such a poset A, the Scott topology S has basis { k k  k(A)} and the Lawson topology has subbasis S@{A¯ k k  k(A)} (see [7], Corollary II.1.15 and Definition III.1.5 as well as [11], §V). An element in a lattice L is join-irreducible if a"0L (supposing the latter exists) and for all b, c L, a =b–c implies a=b or a= c; denote the set of join-irreducibles by J(L). Let P be a poset. If Q ¤P, let min Q be the least element of Q (if it exists). Let Min Q denote the set of minimal elements of Q and Max Q the set of maximal elements. A poset P has the ascending chain condition (ACC) if whenever p0 5 p1 5 · · · is a chain in P, there exists n N0 such that pn = pn + 1 = · · · . Dually, define the descending chain condition (DCC).

J. D. FARLEY

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Let P and Q be ordered topological spaces. Let P Q t be the poset of continuous order-preserving maps from Q to P (the latter being endowed with the topology t) Q be P Q ordered pointwise: i.e., for f, g  P Q t , f5 g if f(q)5 g(q) for all qQ. Let P t where t is the discrete topology on P. If p P, let p; be the constant map into {p}. We shall often identify 2Q with D(Q). If C is a category and X, Y C, then C(X, Y) will be the set of C-morphisms from X to Y. Let Slat denote the category of semilattices with semilattice homomorphisms. If S, T Slat, let Slatfin(S, T) œ {g  Slat(S, T) Im g is finite}. Let DSlat denote the class of distributi6e –-semilattices, the –-semilattices S such that for all a, b, c S, if a 5b–c then there exist b %5 b and c %5 c such that a= b %–c %. A –-semilattice S is a distributive –-semilattice if and only if S s is a distributive lattice ([8], Lemma II.5.1 (iii)). Let D denote the category of bounded distributive lattices with {0, 1}-preserving homomorphisms. Let P be a poset; let (S, –, 0S )Slat and let i: P“S be an order-embedding. We say (S, i) is the free {0}-–-semilattice generated by P, denoted FSlat(P), if for every (T, –, 0T )  Slat and f  T P, there exists a unique map f( Slat(S, T) such that f( $ i = f. The ordered space P is totally order-disconnected if, for all p, qP such that p5 ⁄ q, there exists U  D(P) such that pU and qQU. It is a Priestley space if it is compact and totally order-disconnected. The category of Priestley spaces with continuous order-preserving maps is denoted P. Let L  D. Denote by P(L) the Priestley space of prime filters, with the appropriate topology. For a  L, let rL (a) œ {F  P(L) a  F}. The functors D and P yield a dual equivalence between D and P. For basic consequences of Priestley duality, see [14]. If tb is a topology on a set Xb (bB), Pb  B tb denotes the product topology on Pb  B Xb. A bitopological space is a triple (X, A, B) where X is a set and A and B topologies on X. Let (X, A, B) and (Y, C, D) be bitopological spaces. A map f: (X, A, B) “(Y, C, D) is continuous if f: (X, A) “ (Y, C) and f: (X, B) “ (Y, D) are continuous. Let (Q, 5 , t) be an ordered space. The upper topology U(Q, 5 , t) is the family of open up-sets (also denoted UQ or U); the lower topology is defined dually (denoted LQ or L). One gets the bitopological space (Q, L, U). Typically a poset Q will be regarded as an ordered space (Q, 5, tdisc), where tdisc is the discrete topology.

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Let (X, A, B) be a bitopological space. Let U¤ X. The pairwise closure cl U of U is the closure of U in (X, A–B), where A–B is the smallest topology containing A and B. The space (X, A, B) is pairwise T2 if, for all distinct x, yX, there exist disjoint U A and V  B such that either xU and yV, or xV and yU. The space (X, A, B) is pairwise completely 2-regular if, for all A-closed C and all x  X¯C, there exists a continuous map f: (X, A, B) “2 such that f(x) =0 and f[C]¤ {1}, and for all B-closed D and all yX¯D, there exists a continuous map g: (X, A, B) “ 2 such that g(y) = 1 and g[D] ¤{0}. The space (X, A, B) is pairwise 2-Tychonoff if it is pairwise T2 and pairwise completely 2-regular. For such a space, let eX : (X, A, B) “ 5 (2, L, U)f f  2X

be defined by eX (x) œ ( f(x))f  2X (xX). Let (b2X, A. , B. ) be cl(Im eX ) with the subspace topologies; with the map . , B. ), eX : (X, A, B) “(b2X, A it is called the Stone-C& ech 2-compactification of (X, A, B).

3. The ideal semilattice of a Priestley power of a semilattice s

) In this section, we show that the isomorphisms (S P(T))s $ (S s)P(T) $ (S s)P(T S L hold, where (S, –)  Slat and TD (Theorem 6 and Corollary 7). We prove that s) s s) s (S s)P(T $(S s)P(T ) if and only if (S s)P(T = (S s)P(T ) (Proposition 26). We establish L L necessary conditions for the isomorphism to hold as well as sufficient conditions; they are necessary and sufficient if SDSlat (Theorems 27 and 28). We characterize L D such that L s satisfies DCC (one of the conditions) in Proposition 25. We also deduce a theorem of Gra¨tzer and Schmidt (Corollary 29; see [9, Theorem 3]).

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The main difficulty with which we must contend is that our –-semilattices need not have a least element. THEOREM 3.1. Let (S, –) Slat, T D. We regard T and T s as the objects (T, ‚ ) and (T s, S ) of Slat, respecti6ely. Let 8: k(T s)$ T be the isomorphism 8(¡t) = t (t  T). Define a map “ C: (S s)P(T) S

!

"

n

g  Slat(S, T s) 0 ti =1T for some nN0, si S, ti g(si ) (i= 1, . . . , n) i=1

as follows: for f (S s)P(T) and s S, let S [C( f )](s) œ{t  T s  Sf[rT (t)]}. Define a map

!

n

F: g  Slat(S, T s) 0 ti =1T for some nN0, si S, ti g(si ) (i= 1, . . . , n)

"

i=1

“(S s)P(T) S as follows: for g Slat(S, T s) such that 0ni = 1 ti = 1T for some nN0, si S, ti g(si ) (i = 1, . . . , n) and F P(T), let [F(g)](F) œ {s S F Sg(s) " ®}. Then C and F are mutually-in6erse order-isomorphisms. Define “ C %: (S s)P(T) L

!

n

g  Slat(S, T) 0 g(si ) = 1T for some nN0, si S (i= 1, . . . , n) i=1

as follows: for f (S s)P(T) and s S, let L

"

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[C %( f )](s) œ 8[(C( f ))(s)]. Define

!

n

F %: g  Slat(S, T) 0 g(si ) =1T for some nN0, si S (i= 1, . . . , n)

"

i=1

“(S s)P(T) L as follows: for g  Slat(S, T) such that 0ni = 1 g(si )= 1T for some n N0, si S (i= 1, . . . , n) and F P(T), let [F %(g)](F) œ g − 1(F). Then C % and F % are mutually-in6erse order-isomorphisms. The restriction of C % to (S s)P(T) maps onto

!

"

g  Slatfin(S, T) 0 Im g = 1T .

Proof. The result follows from [5], Corollary 3.7 by adjoining a least element to S. For I  T s, let rT (I) œ {rT (a) aI}. We have {g  Slat(S, T s) for all F P(T) there exists sS such that g(s)SF" ®} ={g  Slat(S, T s) {rT (g(s)) sS} covers P(T)} = {g  Slat(S, T s) for some nN0, si  S, {rT (g(si )) i= 1, . . . , n} covers P(T)}

! !

n

= g  Slat(S, T s) for some nN0, si S(i= 1, . . . , n), 0 g(si )= T i = 1T s n

"

"

= g  Slat(S, T s) 0 ti =1T for some nN0, si S, ti g(si ) (i= 1, . . . , n) . i=1

LEMMA 3.2. Let (S, –)  Slat and let PP. Let D¤ (S s)PS be a directed subset. Define c: P “ S s for all p P by c(p)œ  d  Dd(p). Then c (S s)PS and c=  D.

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Proof. Clearly c is well-defined and order-preserving. Let s  S. Then c − 1( S s (¡s)) = {p  P s  c(p)} ={p  P s d(p) for some dD} = {p  P p  d − 1( S s (¡s)) for some dD} = . d − 1( S s (¡s))U(P). dD

Hence c (S s)PS. LEMMA 3.3. Let (S, –)  Slat and let PP. Let f(S s)PS and p0 P; suppose s0  f(p0). Then there exists h (S s)P such that Im h¤ k(S s), h5f, and s0 h(p0). Proof. For all p  P, choose sp f(p). Then pf − 1( S s (¡ sp ))U(P), so there exists Up  D(P) such that p  Up ¤f − 1( S s (¡ sp )). By compactness, n

P = . Upi i=1

for some n  N0 and p1, . . . , pn  P. Let si =spi

and

Ui =Upi

(i = 0, . . . , n).

Thus, for all p  P, si  f(p) for some i (15 i5 n). Define h: P “S s for all p  P as follows: h(p) œ ¡ 0 {si p  Ui (i =0, . . . , n)} Then h is order-preserving, Im h is finite, and, for all sS, h − 1( S s (¡ s))  D(P). By [5], Lemma 3.5, h  (S s)P. Clearly h5 f. COROLLARY 3.4. Let (S, –) Slat and let PP. Then for all f (S s)PS, f = {h(S s)P h5 f and Im h¤ k(S s)}. Proof. The result follows from Lemmas 2 and 3.

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LEMMA 3.5. Let (S, –)  Slat and let P  P. Let h(S s)P be such that Im h¤ k(S s). Then h k[(S s)PS]. Proof. Let D¤ (S s)PS be directed and assume h5 cœ D. For all p P, there exists dp  D such that h(p) ¤ dp (p), and hence there exists an open subset Up ¤ P such that p  Up and h(u) ¤ dp (u) for all uUp. By compactness, P= ni = 1 Upi for some n  N0 and p1, . . . , pn  P. Hence there exists dD such that h5 d. THEOREM 3.6. Let (S, –) Slat and let PP. Define F: (S P)s “ (S s)PS as follows: for B (S P)s and p P, let [F(B)](p)œ ¡{b(p) bB}. Then F is an order-isomorphism. Proof. The theorem follows easily from Lemma 2, Corollary 4, Lemma 5, and [4], Proposition 3. COROLLARY 3.7. Let (S, –) Slat and let TD. Then s

) (S P(T))s $(S s)P(T . L

Proof. The corollary follows from Theorems 1 and 6. Now we investigate when we may dispense with the Lawson topology in the above result. The first lemma is easy. LEMMA 3.8. Let (S, –)  Slat and let PP. Assume e6ery non-empty subset of S P has a supremum. Then P"® implies e6ery non-empty subset of S has a supremum. LEMMA 3.9. Let (S, –) Slat and let T D. Assume e6ery non-empty subset of S P(T) has a supremum. Then S satisfies ACC or T satisfies DCC. Proof. Let P œP(T). For a contradiction, assume there exist chains s0 B s1 B · · · in S and U0 U1 · · · in D(P). Define fn : P“ S for all nBv as follows: fn (p) œ

!

sn so

if p  U n, otherwise

for all p  P. Then fn  S P for all nB v.

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Let fœ 0n B v fn and sv œ0n B v sn (which exists by Lemma 8). For all nB v, let gn : P “ S be defined by gn (p) œ

!

sv sn

if p  U n, otherwise

for all p  P. Then gn  S P for all nB v. For all m, n Bv, fm 5gn. Hence, for all nB v, f5gn. Thus, for all nB v, p P¯Un implies f(p) 5sn. For all n Bv, choose pn  Un ¯Un + 1; then sn 5 f(pn )5 sn + 1. Hence Im f is infinite, a contradiction. LEMMA 3.10. Let (S, –)  Slat and let TD. Assume e6ery non-empty subset of S P(T) has a supremum. Then at least one of the following holds: (1) for all s S,  s satisfies DCC; (2) T satisfies ACC. Proof. Let P œ P(T). For a contradiction, assume there exist chains s0 \ s1 \ · · · \ sv in S and U0 àU1 à· · · in D(P). Define fn : P“ S for all nB v as follows: fn (p) œ

!

s0 sn

if p  U n, otherwise

for all p  P. Then fn  S P for all nB v. As s; v  { fn n Bv}l, f œ /n B v fn exists in S P. For all nBv, let gn : P“ S be defined by gn (p) œ

!

sn sv

if p  U n, otherwise

for all p  P. Then gn  S P for all nB v. For all m, n Bv, fm ]gn. Hence, for all nB v, f] gn. Thus, for all n B v, p Un implies f(p)] sn. For all n Bv, choose pn  Un + 1¯Un ; then sn ] f(p)] sn + 1. Hence Im f is infinite, a contradiction. LEMMA 3.11. Let (S, –)  Slat and let T D. Let f (S s)P(T) L . Assume that S satisfies ACC or T satisfies DCC. Then:

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(1) for all I S s, there exists sI such that f − 1( S s {I}) = f − 1( S s {¡ s}) D(P(T)); (2) if in addition T satisfies ACC, then Im f satisfies DCC. Proof. For I  S s, { f − 1( S s{¡ s}) sI} is a filtered subfamily of D(P(T)) whose intersection is f − 1( S s{I}). Now assume T satisfies ACC. Let I0  I1  · · · be a chain in Im f. Then ( f − 1( S s {Ik }))k E 0 is a strictly ascending chain in D(P(T)) by (1). LEMMA 3.12. Let (S, –)  Slat and let PP. Let f(S s)PL. Assume that  S s {I} satisfies DCC for all I S s. Then Im f satisfies DCC. Proof. As P =  { f −1( S s {¡s}) sS}, by compactness there exist nN0, s1, . . . , sn  S such that, for all p P, si f(p) for some i between 1 and n. Hence, for any chain I0 I1 · · · in Im f, for some i between 1 and n, si Ik for k] 0, a contradiction. s

LEMMA 3.13. Let (S, –)  Slat and let TD. Assume that (S P(T))s $ (S s)P(T ). s) Let D¤ (S s)P(T be non-empty and set cœ 0 D. Assume there exists L s fD u S (S s)P(T ) such that c" f. s Then there exists g D u S (S s)P(T ) such that f5 ⁄ g. Proof. Let P œP(T s). By Lemmas 9, 10, 11, and 12, Im c satisfies DCC. By [5], Lemma 3.5, we may assume Im c is infinite. Let J be a minimal element of Im c¯Im f. For all I  Im c and I %  Im f, let VI,I % œ c − 1( S s {I})Sf − 1( S s {I %})D(P); let

HI,I % œ

!

I I%

if I ¤ J, otherwise.

Define g: P “ S s for all p  P as follows:

g(p) œ 0 {HI,I % I Im c, I % Im f, and pVI,I %}. Ss

Then g  (S s)PL and Im g is finite, so g(S s)P ([5], Lemma 3.5). Clearly c 5g; also J  Im g and f 5 ⁄ g.

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) If (S, –)  Slat and T  D and (S s)P(T $ (S s)P(T ), we have seen that we may L deduce certain information about the structure of S s or T s. In the following lemmas, we show how to describe the properties that T s may have directly in terms of T. We start with a trivial statement.

LEMMA 3.14. Let L be a lattice with DCC. Then e6ery element is a finite join of join-irreducibles. Compare the following with [13], Proposition 3.3. LEMMA 3.15. Let L D be such that e6ery element is a finite join of join-irreducibles. Then L$ FSlat (J(L)). Proof. Let P œ J(L). Assume (S, –, 0S )Slat and fS P. Define g: L “S by g(a) =f(a1)–· · · –f(an ) where nN0, a1, . . . , an J(L), and a= a1–· · · –an. Then g is well-defined and g  Slat(L, S). COROLLARY 3.16. Let LD be such that L s satisfies DCC. Then: (1) L $FSlat(J(L)); (2) J(L) has no infinite antichains; (3) Max(J(L)) is cofinal in J(L). Proof. By (1) and [6], Lemma 4.1, L s $ O(J(L)). Assume for a contradiction that {an n ]0} is an infinite antichain in J(L)). Then {¡J(L){ak n 5k} n ] 0} is a strictly descending chain in O(J(L)), a contradiction. Hence (2) holds. By (1), 1L corresponds to ¡J(L)R for some finite antichain R¤ J(L). Hence, for any aJ(L), there exists r  R such that a5r. That is, R is cofinal in J(L). Clearly R=Max(J(L)), so (3) holds. Concerning the following (a simple consequence of Ramsey’s Theorem, [15], Theorem A), see [12], §3. LEMMA 3.17. If (pn )n B v is a family in a poset with no infinite antichains, there exist n0 Bn1 B· · · Bv such that

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or pn 0 \pn 1 \ · · · .

LEMMA 3.18. Let P be a poset satisfying DCC with no infinite antichains. Then O(P) satisfies DCC. Proof. Assume for a contradiction that there exists a chain X0  X1  · · · in O(P). For each n ] 0, choose xn  Xn ¯Xn + 1. By Lemma 17, there exist m, n] 0 such that m B n and xm 5xn. Hence xm Xn ¤ Xm + 1, a contradiction. LEMMA 3.19. Let L  D be such that L s satisfies DCC. Then L satisfies DCC and has no infinite antichains. Proof. By Corollary 16, L $FSlat(P) where P is a poset satisfying DCC with no infinite antichains and such that Max P is cofinal in P. For a contradiction, assume there exist finite antichains R0, R1, . . . , of P such that {¡Rn n ]0} is an infinite antichain of FSlat(P). ⁄ rn for all rn Rn. For all distinct m, n ] 0, there exists am,n Rm such that am,n 5 Therefore, for all m ] 0, there exists am Rm such that {n] 0 am 5 ⁄ rn for all rn  Rn } is infinite. Hence there exist n0 B n1 B · · · in N0 such that 05 iB j implies / anj , ani 5 contradicting Lemma 17. LEMMA 3.20. Let (S, –)  DSlat. Assume that S satisfies ACC and for all s S,  s satisfies DCC. Then for all s S,  s is finite. Proof. For all s S,  s  D satisfies ACC and DCC so is finite (see, for instance, [2], Exercise 10.13). The next lemma is easy. LEMMA 3.21. Let (S, –) Slat and let T D. If  s is finite for all sS, then s s (S s)PL (T ) =(S s)P(T ). LEMMA 3.22. Let (S, –)  Slat and let TD. Assume that S and T satisfy ACC. s) s Then (S s)P(T =(S s)P(T ). L s

) Proof. Let f  (S s)P(T . Let IS s. Then S s¯¡Ss {I}=  s  S¯I  S s{¡s}. Hence L s s f (S ¯¡Ss {I})  U(P(T )) = D(P(T s)) so that f − 1(¡Ss {I}) is clopen. By Lemma 11 (1), f − 1( Ss {I}) is clopen, so f −1({I}) is clopen. −1

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J. D. FARLEY

ALGEBRA UNIVERS.

LEMMA 3.23. Let P be a poset with ACC and no infinite antichains such that Min P is coinitial. Let Q be a poset such that, for all qQ,  q satisfies DCC. Then for all f Q P, Im f is finite. Proof. Clearly Im f has no infinite antichains. Assume for a contradiction that there exists a chain q0 \ q1 \ · · · in Im f. Then there exists m Min P such that f(m){qn n] 0}l, a contradiction. Hence Im f satisfies DCC. Assume for a contradiction that there exists a family (pn )n ] 0 in P such that f(p0) B f(p1) B · · · . By Lemma 17, there exists m, nN0 such that mBn, pm \ pn, and f(pm ) ]f(pn ), a contradiction. Hence Im f satisfies ACC. By Lemma 17, Im f is finite. COROLLARY 3.24. Let (S, –) Slat and let TD. Assume that for all IS s,  S s{I} satisfies DCC and that T s satisfies DCC. s) s Then (S s)P(T =(S s)P(T ). L Proof. By [6], Lemma 4.1 and Corollary 16, T s $ 2P where P is a poset with no infinite antichains satisfying ACC and such that Min P is coinitial. As P(T s) $ b2(P) ([6], Theorem 3.17), Lemma 23 implies Im f is finite for all s) s) s f(S s)P(T . By [5], Lemma 3.5, (S s)P(T = (S s)P(T ). L L PROPOSITION 3.25. Let LD. Then the following are equi6alent: (1) L s satisfies DCC; (2) L satisfies DCC and has no infinite antichains. Proof. Lemma 19 states that (1) implies (2). If (2) holds, then by Lemmas 14 and 15, L $ FSlat(J(L)) where J(L) has no infinite antichains. By Lemma 18, L s satisfies DCC. PROPOSITION 3.26. Let (S, –) Slat and let T D. Then s

s)

) (S s)P(T $(S s)P(T L

if and only if s

s

s

) (S s)P(T = (S s)P(T ). L

s

s

) Proof. Assume (S s)P(T $(S s)P(T ). By Lemma 13, (S s)P(T ) is closed under L s) s P(Ts) non-empty joins in (S )L . Theorem 1 and Corollary 4 imply that (S s)P(T = L s s P(T ) (S ) .

THEOREM 3.27. Let (S, –) Slat and let TD. Assume s

(S P(T))s $(S s)P(T ).

Vol. 41, 1999

Ideals of Priestley powers of semilattices

253

Then at least one of the following holds: (1) T is finite; (2) S and T satisfy ACC; (3) for all I S s,  S s{I} satisfies DCC and T s satisfies DCC; (4) S satisfies ACC and for all sS,  s satisfies DCC. If (S, –)  DSlat, then (4) is equi6alent to (5) for all s S,  s is finite. Proof. If T s satisfies ACC and DCC, then T s is finite (see [2], Exercise 10.13). The theorem follows from Lemmas 9, 10, and 20. s

THEOREM 3.28. Let (S, –) Slat and let TD. Then (S P(T))s $ (S s)P(T ) if at least one of the following holds: (1) T is finite; (2) S and T satisfy ACC; (3) for all I S s,  S s {I} satisfies DCC and T s satisfies DCC; (4) for all s S,  s is finite. [Remark: (4) is equivalent to (5) S satisfies ACC and for all sS,  s satisfies DCC and has no infinite antichains.] Proof. Use Corollary 7. Parts (1) and (4) are obvious; (2) follows from Lemma 22; (3) follows from Corollary 24. COROLLARY 3.29 ([9], Theorem 3). Let L be a lattice and let MD. Then Con(L P(M)) $(Con L)P(Con M) if and only if either Con L is finite or M is finite. M) and by [5], Lemma Proof. By [5], Corollary 6.11, Con(L P(M))$ (Con L)P(Con L s 6.9, Con M$ B where B is the minimal Boolean extension of M. Hence if Con L is finite or M is finite then Con(L P(M))$ (Con L)P(Con M). Now assume Con(L P(M)) $(Con L)P(Con M). Hence s

s

) (Con L)P(B $(Con L)P(B ). L

Obviously B is finite if and only if B satisfies ACC or DCC (see [2], Exercise 10.13). Recall that Con L  D ([1], Theorem II.9.15). By Theorem 27, either B is finite (if (1), (2), or (3) holds) or Con L is finite (if (5) holds).

254

J. D. FARLEY

ALGEBRA UNIVERS.

REFERENCES [1] BALBES, R. and DWINGER, P., Distributi6e Lattices, University of Missouri Press, Columbia, Missouri, 1974. [2] DAVEY, B. A. and PRIESTLEY, H. A., Introduction to Lattices and Order, Cambridge University Press, Cambridge, 1990. [3] DUFFUS, D., JO´NSSON, B. and RIVAL, I., Structure results for function lattices, Canadian Journal of Mathematics 30 (1978), 392–400. [4] ERNE´, M., Compact generation in partially ordered sets, Journal of the Australian Mathematical Society 42 (1987), 69–83. [5] FARLEY, J. D., Priestley powers of lattices and their congruences: a problem of E. T. Schmidt, Acta Scientiarum Mathematicarum 62 (1996), 3 – 45. [6] FARLEY, J. D., Priestley duality for order-preser6ing maps into distributi6e lattices, Order 13 (1996), 65–98. [7] GIERZ, G., HOFMANN, K. H., KEIMEL, K., LAWSON, J. D., MISLOVE, M. and SCOTT, D. S., A Compendium of Continuous Lattices, Springer-Verlag, Berlin, 1980. [8] GRA8 TZER, G., General Lattice Theory, Academic Press, New York, 1978. [9] GRA¨TZER, G. and SCHMIDT, E. T., Congruence lattices of function lattices, Order 11 (1994), 211–220. [10] JO´NSSON, B. and MCKENZIE, R., Powers of partially ordered sets: cancellation and refinement properties, Mathematica Scandinavica 51 (1982), 87 – 120. [11] LAWSON, J. D., The 6ersatile continuous order, Lecture Notes in Computer Science 298 (1988), Springer-Verlag, Berlin, 134–160. [12] Li, BOYU and MILNER, E. C., A chain complete poset with no infinite antichain has a finite core, Order 10 (1993), 55–63. [13] PRIESTLEY, H. A., Catalytic distributi6e lattices and compact zero-dimensional topological lattices, Algebra Univers. 19 (1984), 322–329. [14] PRIESTLEY, H. A., Ordered sets and duality for distributi6e lattices, Annals of Discrete Mathematics 23 (1984), North-Holland, Amsterdam, 39 – 60. [15] RAMSEY, F. P., On a problem of formal logic, Proceedings of the London Mathematical Society 30 (1930), 264–286. J. D. Farley Mathematical Institute Uni6ersity of Oxford 24 – 29 St. Giles’ Oxford OX1 3LB United Kingdom and Department of Mathematics Vanderbilt Uni6ersity Nash6ille, TN 37240 U.S.A. e-mail: [email protected]