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Zhani To the memory of the first author’s mother Abstract. We characterize upper semi-lattice algebras and study their relationship with other classes of Tail algebras. Using the notion of support of nonzero elements, we derive some finite combinatorics on lengths of elements within this class of algebras. Mathematics Subject Classification (2000). Primary: 03G, 06E. Secondary: 06A12, 03G10. Keywords. Tail algebra, semi-group, upper semi-lattice algebra, length of elements. 1. Introduction Let (T, <) be a partially ordered set and consider the subalgebra B(T ) of the power set of T , P(T ) generated by {bt : t ∈ T }, where bt := {x ∈ T : t ≤ x}.

12 applied to Mα we arrive to a contradiction. Suppose now there is S finite such that S Ω ϕ. 12. The Ω-conjecture says: If there exists a proper class of Woodin cardinals, then for each sentence of the language of set theory ϕ, ∅ Ω ϕ iff ∅ Ω ϕ. The “if” direction is given by Soundness. , if ∅ Ω ϕ, then ∅ Ω ϕ, for every ϕ ∈ Sent. 6. The following are equivalent: i) For all ϕ ∈ Sent, ∅ Ω ϕ implies ∅ Ω ϕ. e. set T ∪ {ϕ} ⊆ Sent, T Ω ϕ implies T Ω ϕ. Proof. i) ⇒ ii). e. and ϕ such that T Ω ϕ. Let ϕ∗ := “T Ω ϕ”.

First, f (x0 ) = 0 and f (−x1 ) = −f (x1 ) = −1 = 0, so both x0 and −x1 are in the kernel. Denote the meet operation of B by ∧ to fit into the notation above. If a, b ∈ M , then f (xa · xb xa∧b ) = (a ∧ b) (a ∧ b) = 0 as desired. So I is a subset of the kernel of f . Now suppose that u ∈ A is in the kernel of f . Write xai1 · · · xaipi · −xbi1 · · · − xbiqi , u= i

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1973, Year of the Humanoids: An Analysis of the Fall UFO Humanoid Wave by David Webb

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