IIteratively algebraic orders
P.H. Rodenburg
Institute for Informatics, University of Amsterdam
A short proof of a theorem of M.H. Albert, and its application to lattices.
Introduction
In a preliminary version of [1], Grätzer asked if there exists a lattice that is isomorphic to its lattice of ideals and in which not every ideal is principal. This question was answered in the negative by D. Higgs [2]. A related question was posed by H.-E. Hoffmann in [3]: whether an alge-braic order (poset) whose compact elements again form an algebraic order and so on, can have noncompact elements. It was answered in the negative by M.H. Albert [4]. Having forgotten a crucial element of Higgs’ proof, and unable to under-stand the proof in [4], I invented a simpler proof, the second, and hopefully this time correct, version of which I present below, after a brief rehearsal of definitions. At the end I will indicate what baffled me.
Substance
This note is about ordered sets, that is, sets with a (partial) ordering rela-tion ≤ on them. We write x < y if x ≤ y and x ≠ y , and x -‹ y if y is an upper cover of x , that is, x < y and if x ≤ z ≤ y then z is either x or y . A chain is a linearly ordered set. Definition 1 . A subset X of an ordered set L is directed if every finite subset of X has an upper bound in X . In particular, the void subset of a directed set has an upper bound, so directed sets are nonvoid. Definition 2 . An ordered set L is upwards complete if every directed X L has a supremum in L . The supremum of X is denoted by \/ X . We write ( X ] L , omitting the sub-script if it can be derived from the context, for { y L | for some x X , y ≤ x }. Instead of ({ x }], we write ( x ]. Dually we have [ X ) and [ x ). Definition 3 . An element k of an ordered set L is compact if for every di-rected X L , k ≤ \/ X implies k ( X ]. We denote the set of compact elements of an ordered set L by K ( L ). We put K ( L ) = L , K n +1 ( L ) = K ( K n ( L )). 2 Definition 4 . An ordered set L is algebraic if it is upwards complete and for every x L , ( x ] K ( L ) is directed and x is its supremum. It is iteratively al-gebraic if for all n , K n ( L ) is algebraic. Theorem . If an ordered set A is iteratively algebraic, A = K ( A ). Proof . Assume A is iteratively algebraic, and a A – K ( A ). Since the supre-mum of a chain of noncompact elements is noncompact, by Zorn’s Lemma, A contains a maximal noncompact element m ≥ a . Clearly, [ m ) satisfies the ACC — the supremum of an infinitely ascending chain cannot be com-pact. The element m is the supremum of a directed set C of compact ele-ments. We know that m is not in K ( A ); but C K ( A ), and since K ( A ) is al-gebraic, C has a supremum a in K ( A ). Now a is noncompact in K ( A ). Again using Zorn’s Lemma, we find a maximal noncompact m ≥ a in K ( A ). Then m ≥ a > m ; and repeating the argument we get an infinitely ascending chain m < m < m <… in [ m ), contradicting the ACC. So A = K ( A ). Corollary . If a lattice L is isomorphic to its ideal lattice, all its ideals are principal. Proof . As an ordered set, the ideal lattice Idl( L ) is algebraic; so likewise L is algebraic. ( L will even be an algebraic lattice if it has a 0.) The compact ele-ments of Idl( L ) are the principal ideals. The sublattice of principal ideals is obviously isomorphic to L , so L K ( L ), which implies that L is iteratively algebraic. Then by the Theorem, all the elements of L , and hence all the ele-ments of Idl( L ), are compact. Discussion
The proof of the theorem certainly owes to Higgs, but omits his main idea: a construction of double sequences of compact elements, based on the observation that a lower cover of an ideal generated by a compact element must be principal. Albert [4] claims to prove the theorem, but his conclusion that A = K ( A ), after a transfinite lopping off of maximal elements, appears right out of the blue. Hansoul [5] suggests a proof of the dual of Albert’s theorem along the lines of [2]. References [1] G. Grätzer,
General lattice theory . Basel 1978. [2] Denis Higgs,
Lattices isomorphic to their ideal lattices . Algebra Universalis 1 (1973), 71-72. Continuous posets and adjoint sequences . Semi-group Forum 18 (1979), 173-188. [4] M.H. Albert,
Iteratively algebraic posets have the ACC . Semigroup Forum 30 (1984), 371-373. [5] Georges Hansoul,