A characterization of well-founced algebraic lattices


  • Maurice Pouzet
  • Ilham Chakir




We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements.  More specifically, we show that an algebraic lattice L is well-founded if and only if K(L), the join semi-lattice of compact elements of L, is well founded and contains  neither [\omega]^\omega, nor \underscore(\Omega)(\omega*) as a join semilattice.  As an immediate corollary, we get that an algebraic modular lattice L is well-founded if and only if K(L) is well founded and contains no infinite independent set.  If K(L) is a join-subsemilattice of I_{<\omega}(Q), the set of finitely generated initial segments of a well founded poset Q, then L is well-founded if and only if K(L) is well-quasi-ordered.


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