Statement:
BPI: Every Boolean algebra has a prime ideal.
Howard_Rubin_Number: 14
Parameter(s): This form does not depend on parameters
This form's transferability is: Transferable
This form's negation transferability is: Negation Transferable
Article Citations:
Stone-1936: The theory of representations for Boolean algebras
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 14 A | Ultrafilter Theorem: Every proper filter over aset \(S\) (in \({\Cal P}(S)\)) can be extended to an ultrafilter.Jech [1973b] p 17. |
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| 14 B | Stone Representation Theorem: Every Boolean algebrais isomorphic to a set algebra. Jech [1973b] p 27 prob 2.12 andM.~Stone [1936]. |
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| 14 C | Every commutative ring with unit has a prime ideal.Scott [1954]. |
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| 14 D | Every ring with unit has a (two sided) prime ideal.(An ideal \(P\) in a ring \(R\) is prime if \(P \neq R\) and for anyideals \(A\) and \(B\) in \(R\), if \(AB \subseteq P\) then \(A \subseteq P\) or\(B\subseteq P\).) Blass [1986] and Banaschewski [1985]. |
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| 14 E | Almost Maximal Ideal Theorem: Every bounded,non-trivial, distributive lattice has an almost maximal ideal.(Definitions: If \(D\) is a bounded non-trivial distributivelattice, \(I(D)\) is the lattice of ideals of \(D\). An almost maximalideal in \(D\) is an ideal \(P\) of \(D\) such that \(s(P) = P\) where \(s\) isthe operation on \(I(D)\) defined as follows: If \(e\) is the unit of \(I(D)\),(i) For \(a\in I(D)\) we say \(x\in I(D)\) is \(a\)-small if \(x\vee b = e\)implies \(a\vee b = e\) for all \(b \in I(D)\). (ii) For any \(a\in I(D)\),\(s(a)\) is the join of all a-small elements of \(I(D)\)).Banaschewski/Harting [1985] and Blass [1986]. |
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| 14 F | Every non-trivial complete distributive lattice withcompact unit has a prime ideal. Banaschewski [1985] and Note 29. |
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| 14 G(n) | (\(n\in \omega\), \(n\ge 3\)) \(P_{n}\): Forevery graph \(G\) if every finite subgraph of \(G\) is \(n\)-colorablethen \(G\) is \(n\)-colorable. L\"auchli [1971] and Note 117. |
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| 14 H | Compactness Theorem for First Order Logic:If \(\Sigma\) is a set of formulas in a first order language such thatevery finite subset of \(\Sigma\) has a model then \(\Sigma\) has a model .Henkin [1954a] and Jech [1973b] p 17. |
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| 14 I | Consistency Principle: For every binary mess \(M\) ona set \(S\) there is a function \(f\) on \(S\) which is consistent with \(M\).Jech [1973b] p 17 and Note 109. |
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| 14 J | Tychonoff's Compactness Theorem for Compact HausdorffSpaces: The product of compact Hausdorff spaces is compact. (For manycompactness theorems \(T\), it is clear that [14 J] implies \(T\) implies[14 K].) \L o\'s/Ryll-Nardzewski [1954] and Rubin/Scott [1954]. |
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| 14 K | Compactness of the Generalized Cantor Space:\(\{0,1\}^I\) is compact for any I. (For many compactness theorems \(T\), itis clear that [14 J] implies \(T\) implies [14 K].) Jech [1973b]p 28, prob 17 and Mycielski [1964]. |
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| 14 L | Stone-\v Cech Compactification Theorem: If \(X\) isa completely regular, T\(_1\) space and \(f\) is a continuous function from\(X\) to a compact Hausdorf space \(Y\), then there is a continuous extensionof \(f\) which carries the compactification \(\beta(X)\) into \(Y\).Kelley [1955],Rubin/Scott [1954] and Note 71. |
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| 14 M | R. Cowen's Generalization of Konig's Lemma: Let\(T\) be a collection of locally finite trees such that for any finiteset of levels of \(T\), there is a consistent set of verticespiercing those levels. Then there is a consistent set ofvertices piercing the entire set of levels of T. Cowen [1977b]and Note 21. |
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| 14 N(n) | (\(n\in\omega\), \(n\ge 2\)) [14 M] restrictedto families of trees where no tree in the family has a vertex with morethan \(n\) successors. Cowen [1977b], Howard [1984a],and Note 21. |
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| 14 O | Let \(\{ A(i): i\in I\}\) be a collection of finite setsand \(R\) a symmetric binary relation on \(\bigcup^{}_{i\in I}A(i)\). Supposethat for every finite \(W\subseteq I\), there is an \(R\)-consistent choicefunction for \(\{A(i): i\in W\}\). Then there is an \(R\)-consistent choicefunction for \(\{A(i): i\in I\}\). \L o\'s/Ryll-Nardzewski [1951]and Note 109. |
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| 14 P(\(n\)) | (For \(n\ge 3\)) [14 O] restricted to \(\{A(i):i\in I\}\) such that for all \(i\in I\), \(|A(i)|\le n\). Cowen [1977b] and Note 109. (For \(n = 2\) seeForm 141.) |
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| 14 Q | Alaoglu's Theorem: The unit ball is compact in theweak\(^*\) topology in the dual of a Banach space. (See [52 E].)Rubin/Scott [1954] and Note 23. |
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| 14 R | Every coherent local lattice is spatial.Banaschewski [1981] and Note 29. |
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| 14 S | Every compact, regular local lattice is spatial.Banaschewski [1981] and Note 29. |
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| 14 T | Every regular commutative ring with unit has amaximal ideal. Banaschewski [1983] and Note 29. |
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| 14 U | Every closed proper ideal in a Gelfand algebra iscontained in some maximal ideal. Banaschewski [1983] and Note 29. |
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| 14 V | FIP Extension Principle: Let \({\Cal F}\subseteq2^{X}\) have the finite intersection property, then \({\Cal F}\subseteq U\) for some ultrafilter \(U\) on \(X\). Schechter [1996a]. |
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| 14 W | Universal Subnet Theorem: Any net has a universalsubnet. Schechter [1996a] and Note 31. |
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| 14 X | Cowen-Engler Lemma: Let \(\Lambda\) and \(X\) be sets.Let \(\Phi\) be a family of functions from subsets of \(\Lambda\) into X.Assume\itemitem{(a)} \(\Phi(\lambda) = \{f(\lambda ): f\in\Phi\) and \(\lambda\in\)dom \(f\}\) is finite for each \(\lambda\in \Lambda\),\itemitem{(b)} each finite \(S\subseteq\Lambda\) is the domain of atleast one \(f\in\Phi\), and\itemitem{(c)} \(\Phi\) has finite character, (that is, a function \(f\) froma subset of \(\Lambda\) into \(X\) is in \(\Phi\) if and only if every finitesubset of \(f\) is in \(\Phi\)).\item{}Then at least one \(f\in\Phi\) has domain \(\Lambda\). Schechter [1996a]. (See [14 I].) |
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| 14 Y | Form 99 +Form 62. Note 33. |
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| 14 Z | Tychonoff's Compactness Theorem for Families of FiniteSpaces: If \(\{ X_i:i\in I\}\) is a family of finite topological spaces, then\(\prod_{i\in I}X_i\) is compact. (Cleary [14 Z] implies [14 K] and isimplied by [14 J].) Jech [1973b] p 28 prob 17,A.A. Lewis [1983], and Wolk [1967]. |
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| 14 AA | Alexander's Subbase Lemma: A topological space iscompact if and only if there is a subbase \(S\) for the topologysuch that every cover by elements of \(S\) has a finite subcover.Kelley [1955] p 143, A.A. Lewis [1983], Wolk [1967], and Note 34. |
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| 14 AB | Tychonoff's Compactness Theorem for Sober Spaces: Aproduct of compact, sober topological spaces is compact.Blass [1986], Johnstone [1984] and Note 37. |
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| 14 AC | (Depends on \(k\), \(n\in\omega\) with \(k\ge 2\), \(n\ge 2\)and \(n + k\ge 5\).) \(P(k,n)\): If \(X\) is a set and \(P\) is a property ofsubsets of \(X\) of \(n\) character (that is, \(\forall y\subseteq X (P(y)\)iff \(\forall z\subseteq y (|z|\le n\rightarrow P(z))))\), then if everyfinite subset of \(X\) can be partitioned into \(k\) or fewer \(P\)-sets(that is, sets \(z\) such that \(P(z)\)) then \(X\) can be partitionedinto \(k\) or fewer \(P\)-sets. Cowen [1982]. |
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| 14 AD | (Depends on \(k\in\omega\), \(k\ge 2\)) \(P(k)\):If \(X\) is a set and \(P\) is a property of subsets of \(X\) of finite character(that is, \(\forall y\subseteq X (P(y)\) iff \(\forall z\subseteq y (z\)finite \(\rightarrow P(z)))\)), then if every finite subset of \(X\) can bepartitioned into \(k\) or fewer \(P\)-sets (that is,sets \(z\) such that \(P(z))\), then \(X\) can be partitioned into\(k\) or fewer \(P\)-sets. Cowen [1982]. |
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| 14 AE | For any distributive lattice \(D\) with zero and unit,any ideal \(J \subseteq D\) and \(a \not\in J\), there is a prime filter \(F\)such that \(a \in F\) and \(F \cap J = \emptyset\). Banaschewski [1981]. |
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| 14 AF | Disjoint Transversal axiom (2,3): If \(X\) is a set,\({\Cal U}\) is a subset of \(\{y\subseteq X : | y|\le 2\}\) and \({\Cal V}\)is a subset of \(\{y\subseteq X: | y|\le 3\}\) and if for each two finitesubsets \({\Cal U}_{0}\subseteq{\Cal U}\) and \({\Cal V}_{0}\subseteq{\Cal V}\)there are transversals \(Y_{1}\) and \(Y_{2}\) for \({\Cal U}_{0}\) and\({\Cal V}_{0}\) respectively such that \(Y_{1}\cap Y_{2}=\emptyset\), thenthere are disjoint transversals \(X_{1}\) and \(X_{2}\) for \({\Cal U}\) and\({\Cal V}\) respectively. (\(Y\subseteq X\) is a transversal for \({\Cal U}\)if for all \(y\in{\Cal U}\), \(y\cap Y\neq\emptyset\).) Schrijver [1978]. |
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| 14 AG | Topological Rado Lemma: Let \(\{T_i: i\in I\}\)be a family of compact Hausdorff spaces and suppose for each finite\(F\subseteq I\), there is a set \(\Phi(F)\) of partial choice functionson \(I\) whose domains include \(F\) and such that (a) \(F_1\subseteq F_2\)implies \(\Phi(F_2)\subseteq\Phi(F_{1})\) and (b) the set\(\{\phi\vert F: \phi\in\Phi(F)\}\) is closed in \(\prod\{T_i: i\in F\}\) inthe product topology. Then there is a choice function \(f\) on \(I\)satisfying for any finite \(F\subseteq I\), \(\exists\phi\in\Phi(F)\) suchthat \(f\) and \(\phi\) coincide on \(F\). Rav [1977]. |
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| 14 AH | Principle of Consistent Choices: Let \(\{T_i:i\in I\}\) be a family of compact Hausdorf spaces and suppose \(R\) is asymmetric binary relation defined for all pairs of elements belongingto distinct \(T_i\) and such that for all \(i\neq j\) in \(I\), the set\(E(i,j)=\{(x,y): x\mathrel R y\) and \(x\in T_i\) and \(y\in T_j\}\) isclosed in \(T_i\times T_j\). Furthermore suppose that for every finitesubset \(F\subseteq I\), with at least two elements, \(\exists\phi\in\prod\{T_i:i\in F\}\) such that \(\phi(i)\mathrel R\phi(j)\) for all \(i\neq j\)in \(F\). Then \(\exists f\in\prod\{T_i: i\in I\}\) such that \(f(i)\mathrelR f(j)\) for all \(i\neq j\) in \(I\). Rav [1977]. |
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| 14 AI | Engler's Lemma: Let \({\Cal E}\) be a family offunctions whose domains are subsets of a set \(S\) with range \(\subseteq\{0,1\}\) and such that\itemitem{(a)} For every finite \(F\subseteq S\), \(\exists \phi \in {\Cal E}\)with domain F.\itemitem{(b)} The restriction of any function in \({\Cal E}\) to any subsetof its domain is in \({\Cal E}\).\itemitem{(c)}If \(\phi \) is any function with domain \(\phi\subseteq S\) suchthat the restriction of \(\phi \) to any finite subset of its domain is in\({\Cal E}\) then \(\phi \in {\Cal E}\).\item{} Then \(\exists f\in {\Cal E}\) whose domain is \(S\).Rav [1977]. |
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| 14 AJ | Robinson's Valuation Lemma: Let \(\{\phi_{x}:x\in X\}\) be a family of partial valuations on a set \(S\). (\(\phi\) is apartial valuation on \(S\) if the domain of \(\phi\subseteq S\) and therange of \(\phi \subseteq \{ 0,1\}\).) Suppose \(\frak B\) is a filterbase on \(X\) such that for every finite \(F\subseteq S\) and forevery \(B\in\frak B\) there is \(x\in B\) such that \(F\subseteq\) dom\(\phi_{x}\). Then there is a total valuation \(f\) on \(S\) (that is, dom \(f= S\)) such that for all finite \(F\subseteq S\) and all \(B\in\frak B\),\(\exists x\in B\) such that \(F\subseteq\) dom \(\phi_{x}\) and \(f|F = \phi_x|F\). (\(\frak B\) is a filter base means \(\forall B_{1},\ B_{2}\in\frak B\),\(\exists B_{3}\in\frak B\) such that \(B_{3}\subseteq B_{1}\cap B_{2}.)\)Rav [1977] and Cowen [1973]. Compare with [14 I]. |
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| 14 AK | Let \(X\) be a set and \({\Cal D}=\{Y_{i}: i\in I\}\) afamily of finite pairwise disjoint subsets of \({\Cal P}(X)\).Suppose for every finite subset \(\{a, b,\ldots, k\}\) of \(I\),\(\exists s\in I\) such that \(Y_{s}\subseteq Y_{a}\land Y_{b}\land \ldots\land Y_{k}\) (If \(Y\) and \(Z\) are finite pairwise disjoint sets then\(Y\land Z = \{A\cap B: A\in Y\) and \(B\in Z\) and\(A\cap B\neq\emptyset\}\).) Then there is a choice function \(\mu\) whosedomain includes \({\Cal D}\) such that the family \(\{\mu(Y_{i}): i\in I\}\)(where \(\mu(Y_{i})\in Y_{i})\) of subsets of \(X\) is closed under finiteintersections. Rav [1977]. |
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| 14 AL | The radical of a proper ideal \(A\) in a commutativering \(R\) with identity is the intersection of all prime ideals in \(R\)containing \(A\). (The radical of \(A = \{x\in R\) : for some\(m\in\omega, x^{m}\in A\}.)\) Rav [1977] and Note 80. |
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| 14 AM | Let \(A\) be a subring of a commutative ring \(R\) and\(p\) a prime ideal in \(A\) such that \(p = Rp \cap A\). Then there is aprime ideal \(J\) in \(R\) such that \(p = J \cap A\). Rav [1977] andNote 80. |
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| 14 AN | In every commutative ring with identity, everyproper ideal is included in some prime ideal. Rav [1977]. |
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| 14 AO | Suppose \(R\) is a commutative ring, \(A\) is aproper ideal in \(R\), and \(S\) is a multiplicative semigroup in \(R\)not meeting \(A\). Then there is a prime ideal \(p\) in \(R\) such that\(A \subseteq p\) and \(p\cap S=\emptyset\). Rav [1977] and Note 80. |
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| 14 AP | Suppose to each finite subset \(F\) of a set I therecorresponds a set \(\Phi (F)\) of functions whose domains are subsetsof \(I\) including \(F\) and such that (a) \(F_{1}\subseteq F_{2}\) implies\(\Phi(F_{1})\subseteq\Phi(F_{2})\) and (b) \(\forall i\in I\), \(\bigl\{\phi(i): \phi\in\bigcup\{\Phi(F): F\hbox{ finite and }F\subseteq I\}\bigr\}\)is finite. Then there is a function \(f\) with domain \(I\) suchthat for all finite \(F\subseteq I\), \(\exists\phi\in\Phi(F)\)such that \(\phi\) and \(f\) coincide on F. Rav [1977], Cowen [1973] and Note 80. |
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| 14 AQ | Strong Rado Lemma 1: Let \(T\) be a family of finitesets and let \({\Cal F}\) be a collection of finite subsets of \(T\)satisfying\itemitem{(1)} \((\forall K\in T)(\exists S\in{\Cal F})(K\in S)\)\itemitem{(2)} \((\forall S_{1}, S_{2}\in{\Cal F})(\exists S_{3}\in{\Cal F})(S_{1}\cup S_{2}\subseteq S_{3})\)\item{}Assume there is a function \(\phi\) with domain \({\Cal F}\) such thatfor each \(S\in{\Cal F}\), \(\phi(S) = \phi_{S}\) is a function with domain\(S\) and satisfies \((\forall K\in S)(\phi_{S}(K)\in K)\). Then there is afunction \(f\) with domain \(T\) and with the following property: For all\(S\in{\Cal F}\) there is an \(S'\in{\Cal F}\) such that \(S\subseteq S'\) and\(f\) and \(\phi_{S'}\) agree on \(S\). Howard [1993]. |
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| 14 AR | Patching Lemma: Let \(L\) be a local system of \(S\)(that is, \(L\subseteq\Cal P(S)\) and for each finite \(E\subseteq S\),there exists \(H\in L\) such that \(E\subseteq H\)). Let \(F\) be a set and\(n\) a positive integer. Suppose that for each \(H\in L\), there is afunction \(f_{H}: H^{n}\rightarrow F\) and \(\left\{f_{H}(x): H\in L\right\}\)is finite for each \(x\in S^{n}\). Then there is a function \(f:S^{n}\rightarrow F\) such that for any finite subset \(K\subseteq S^{n}\),there is an \(H\in L\) with \(K \subseteq H^{n}\) and \(f\) and \(f_{H}\) agreeon K. Hickin/Plotkin [1976] and Note 83. |
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| 14 AS | Let \(\frak A\) and \(\frak B\) be (universal)algebras of type \(r\) where \(\frak A\) and \(\frak B\) have underlyingsets \(A\) and \(B\) respectively and \(B\) is finite. Suppose forevery finite \(C \subseteq A\) there is a partial \(r\) homomorphismfrom \(\frak A\) into \(\frak B\) with domain \(C\). Then there is an \(r\)homomorphism from \(\frak A\) into \(\frak B\) . van Benthem [1975]. |
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| 14 AT | Assume \(D\) and \(E\) are sets and for all \(d\inD\), \(d\) is finite. Suppose that for every finite \(F\subseteq D\), thereis an \(S\) such that \((\forall f\in F)(f\cap S\in E\)). Then there is an\(S\) such that \((\forall d\in D)(d\cap S\in E)\). van Benthem [1975]. |
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| 14 AU | If \(Z\) is a real vector lattice with order unit 1then the set of all positive linear functionals \(u\) on \(Z\) with \(u(1) =1\) is a convex set that has an extreme point. See H.~Rubin/J.~Rubin [1985], pages 177 and 182 for definitions. Also seeGardiner [1974]. |
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| 14 AV | Every Boolean algebra has a 2-valued measure.Pincus [1972c] and Note 147. |
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| 14 AW | The Compactness Theorem for Propositional Logic:If \(\Sigma\) is a set of formulas in a propositional language such thatevery finite subset of \(\Sigma\) is satisfiable, then \(\Sigma\) issatisfiable. Henkin [1954b]. |
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| 14 AX | Structural Completeness Theorem for PropositionalLogic: Assume \(At\) is a set of propositional variables and \(S\)is the set of propositional formulas generated by \(At\). Assume\(X\subseteq S\) and \(a\in S\) satisfy the following: For any function\(e: At\rightarrow S\) if \(Cn(\{ b[e]: b\in X\})\subseteq Cn(\emptyset)\)then \(a[e]\in Cn(\emptyset)\). Then \(a\in Cn(X)\). (\(Cn(Y)\) is thedeductive closure of \(Y\ \cup\) all substitutions of axioms for theclassical predicate calculus and for \(a\in S\), \(a[e]\) is \(a\) with eachpropositional variable \(p\) replaced by \(e(p)\).) Pogorzelski/Prucnal [1974]. |
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| 14 AY | Let \(F\) be a class of partial functions on \(I\) offinite character such that (a) \(\{f(\nu): f\in F\}\) is finite for each\(\nu\in I\) and (b) for each finite \(W\subseteq I\), there exists \(f\in F\)with dom\((f) = W\). Then \(F\) contains a function with domain \(I\). (\(F\) isof finite character means \(f\in F\) iff for all finite \(w\subseteq I\),\(f|w\in F\).) Cowen [1973]. |
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| 14 AZ | Let \(\Phi = \{\phi_{t}\}_{t\in T}\) be a set ofpartial valuations on a set \(S\) (that is, dom\((\phi_{t})\subseteq S\) andrange\((\phi_{t})\subseteq \{0,1\})\) such that for all finite\(U\subseteq S\) there is a \(t\in T\) for which \(U \subseteq\) dom\((\phi_{t})\).Then there is a valuation \(\Psi\) with dom\((\Psi) = S\) and such that forevery finite \(U\subseteq S\), there is a \(t\in T\) with \(U\subseteq\hbox{dom}(\phi_{t})\) and \(\Psi|U = \phi_{t}| U\). Cowen [1973]. |
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| 14 BA | Monteiro's Theorem for Complete Atomic BooleanAlgebras: Let \(K\) be a complete atomic Boolean algebra,\(A\) a subalgebra of the Boolean algebra \(B\) and \(f\) a homomorphism from\(A\) to \(K\). Let \(d\) be a semimorphism from \(B\) into \(K\) (that is, \(d :B \to K,\ d(0) = 0,\ d(1) = 1\) and for all \(x, y\in B\), \(d(x \lory) = d(x) \lor d(y)\) ) with \(f(x) \le d(x)\) for all \(x \in A\). Thenthere is a homomorphism \(h\) from \(B\) to \(K\) such that \(h(x) \le d(x)\)for all \(x \in B\) and \(h/A = f\). Monteiro [1965] and Bacsich [1972a]. |
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| 14 BB | Fra\"\i ss\'e's Theorem: Let \(\Cal E\) be inductivelyordered (that is, for any \(E,\ E'\in\Cal E,\ \exists E''\in\Cal E\)such that \(E\cup E'\subseteq E''\).) For any \(E\in\Cal E\), assume\(\Cal R_E\) is a finite set of \(m\)-ary relations on \(E\). If for any \(E\),\(E'\in\Cal E\) with \(E\subseteq E'\) we have \(\forall R\in\Cal R_{E'}\),\(R/E \in \Cal R_E\) then there is an \(m\)-ary relation \(S\) on\(\bigcup\Cal E\) such that \(S|E \in\Cal R_E\) for any \(E\in\Cal E\).B\'en\'ejam [1970]. See [14 Y]. |
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| 14 BC | If \(m_0\) is an additive, real valued measure on asubalgebra \(\Cal B_0\) of a Boolean algebra \(\Cal B\) then there existsa measure \(m\) on \(\Cal B\) such that \(m=m_0\) on \(\Cal B_0\) and the rangeof \(m\) is contained in the closure of the range of \(m_0\).Tarski [1930] and Note 147. |
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| 14 BD | Let \(H\) be a hypergraph. If every finite subhypergraphhas an independent vertex cover then \(H\) has an independent vertex cover.Cowen [1990] and Note 117. |
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| 14 BE | Let \(H\) be a hypergraph. If every finite subhypergraphis 2-colorable then \(H\) is 2-colorable. Cowen [1990] and Note 117. |
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| 14 BF | Let \(H\) be a hypergraph whose edges contain at mostthree vertices. If every finite subhypergraph of \(H\) has an independentvertex cover then \(H\) has an independent vertex cover.Cowen [1990] and Note 117. |
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| 14 BG | Let \(H\) be a hypergraph whose edges contain at mostthree vertices. If every finite subhypergraph of \(H\) is 2-colorable then\(H\) is 2-colorable. Cowen [1990] and Note 117. |
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| 14 BH | The sum of finite frames is a spatial frame.Paseka [1989] and Note 29. |
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| 14 BI | Every compact, conjunctive frame is spatial.Paseka [1989] and Note 29. |
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| 14 BJ | The sum of compact, regular frames is spatial.Paseka [1989] and Note 29. |
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| 14 BK | The sum of 4-element Boolean algebras is spatial.Paseka [1989] and Note 29. |
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| 14 BL | Every compact, 0 dimensional frame is spatial.Paseka [1989] and Note 29. |
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| 14 BM | A system of equations over a finitering \(R\) has a solution in \(R\) if and only if every finite sub-systemhas a solution in \(R\). Abian [1972] and Note 30. |
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| 14 BN | A system of equations over a finite field \(F\) has asolution in \(F\) if and only if every finite sub-system has a solutionin \(F\). Cowen/Emerson [1996] and Note 30. |
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| 14 BO | A system of equations over a finite structure\((A,+,\cdot)\) has a solution in \(A\), if and only if every finitesub-system has a solution in \(A\). Note 30. |
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| 14 BP | Every non-empty bounded distributive lattice has aprime ideal. Morillon [1988]. |
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| 14 BQ | Every proper ideal in a bounded distributive latticeis the intersection of the prime ideals containing it. Morillon [1988] and Note 79. (Compare with [14 AE].) |
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| 14 BR | Every \(\lor\)-compact non trivial lattice has aprincipal prime ideal. Morillon [1988], \cite{1985} and Note 71. |
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| 14 BS | Every bounded distributive lattice has a prime idealchoice function. Morillon [1988], \cite{1985} and Note 71. |
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| 14 BT | If \((A_i)_{i\in I}\) is a family of bounded,distributive, \(\lor\)-compact lattices and if \((J_i)_{i\in I}\) is a familysuch that for all \(i\in I\), \(J_i\) is a proper ideal in \(A_i\) then thereis a family \((P_i)_{i\in I}\) such that for all \(i\in I\), \(P_i\) is a primeprincipal ideal in \(A_i\) containing \(J_i\). Morillon [1985],\cite{1988} and Note 71. |
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| 14 BU | If \((A_i)_{i\in I}\) is a family of distributive nontrivial bounded lattices, then there is a family \((f_i)_{i\in I}\) such thateach \(f_i\) is a prime ideal choice function on \(A_i\). Morillon [1985], \cite{1988} and Note 71. |
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| 14 BV | For every set \(E\) there is an effectively\(E\)-adequate ultrafilter. Morillon [1988], \cite{1985} andNote 71. |
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| 14 BW | For every set \(E\) there is an ultrafilter choicefunction on \(E\). Morillon [1985], \cite{1988} and Note 71. |
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| 14 BX | If \((E_i)_{i\in I}\) is a family of sets and \((F_i)_{i\in I}\) is a family such that for each \(i\), \(F_i\) is a filterin \(\Cal P(E_i)\), then there is a family \((U_i)_{i\in I}\) such that foreach \(i\in I\), \(U_i\) is an ultrafilter in \(\Cal P(E_i)\) containing \(F_i\).Morillon [1985], \cite{1988} and Note 71. |
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| 14 BY | For every topological space \(X\), every proper idealin the normed \({\Bbb R}\)-algebra of continuous bounded functions from\(X\) into \(\Bbb R\) is contained in a maximal ideal. Morillon [1988]. |
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| 14 BZ | Every proper ideal in a non-trivial normed\(\Bbb R\)-algebra or \(\Bbb C\)-algebra with identity is contained in amaximal ideal. Morillon [1988]. |
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| 14 CA | If \((A_i)_{i\in I}\) is a family of non trivialcommutative normed algebras with identity then there is a family\((J_i)_{i\in I}\) such that for each \(i\in I\), \(J_i\) is a maximalideal in \(A_i\). Morillon [1988]. |
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| 14 CB | If \(X\) is a topological space and \(I\) is a proper idealin the algebra of all bounded continuous functions from \(X\) to \(\Bbb R\)such that every function in \(I\) vanishes at at least one point of \(X\)then \(I\) is contained in a maximal ideal. Morillon [1988]. |
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| 14 CC | A projective limit of compact topological spaces iscompact. Morillon [1988] and Note 71. |
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| 14 CD | A projective limit of a directed projective system ofcompact topological spaces is compact. Morillon [1988] andNote 71. |
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| 14 CE | A projective limit of a directed projective system ofcompact non-empty topological spaces is non-empty. Morillon [1988] and Note 71. |
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| 14 CF | A projective limit of a directed projective system ofnon-empty sets is non-empty. Morillon [1988] and Note 71. |
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| 14 CG | A projective limit of a directed projective system offinite sets is compact. Morillon [1988] and Note 71. |
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| 14 CH | For every bounded, distributive lattice\((A,\land,\lor)\) in which \(0\) is \(\land\)-compact, for all \(a\in A\),\(a\ne 0\) and for all sequences \((X_n)_{n\in\omega}\) of subsetsof \(A\), there is a prime filter in \(A\) which contains \(a\) and respectseach \(X_n\). Morillon [1988] and Note 71. |
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| 14 CI | 3-Sat: Restricted Compactness Theorem forPropositional Logic II: If \(\Sigma\) is a set of formulas in apropositional language such that every finite subset of \(\Sigma\)is satisfiable and if every formula in \(\Sigma\) is a disjunctionof at most three literals, then \(\Sigma\) is satisfiable.(A literal is a propositional variable or its negation.)Cowen [1990]. |
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| 14 CJ | The Completeness Theorem for Propositional Logic:Every consistent set of formulas is satisfiable. Henkin [1954a]and \cite{1954b}. |
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| 14 CK | The Completeness Theorem for First Order Logic:Every consistent set of formulas is satisfiable. Henkin [1954a]and \cite{1954b}. |
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| 14 CL | Gelfand Extreme Point Theorem: If \(A\) is anon-trivial Gelfand algebra then the closed unit ball in the dualof \(A\) has an extreme point \(e\) with \(e\ne 0\). Morillon [1986],notes 23 and 29. |
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| 14 CM | Kolany's Patching Lemma. Assume that\(\{A_j : j\in J\}\) is a family of non-empty sets and \(\{ \Cal F_j :j\in J\}\) is a family of finite non-empty sets of functions such thatfor every \(j\in J\) and every \(f\in \Cal F_j\), dom \(f = A_j\). Assumethat for every finite \(J_0 \subseteq J\) there is a function \(F_0\) suchthat for all \(j\in J_0\), \(F_0\mid A_j \in \Cal F_j\), then there existsa function \(F\) such that for all \(j\in J\), \(F\mid A_j \in \Cal F_j\).Kolany [1999]. |
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| 14 CN | A system of polynomial equations over the field\(\{0,1\}\) has a solution if and only if every finite subsystem hasa solution. Note 30. |
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| 14 CO | Every \(B\) compact space is compact. Herrlich [1996a] and Note 6. |
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| 14 CP | Every closed filter in a topological space iscontained in a prime closed filter. Herrlich/Steprans [1997] and Note 10. |
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| 14 CQ | Every z-filter in a topological space is containedin a prime z-filter. Herrlich/Steprans [1997] and Note 10. |
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| 14 CR | Every z-filter in a topological space is containedin a maximal z-filter. Herrlich/Steprans [1997] and Note 10. |
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| 14 CS | Every clopen filter in a topological space iscontained in a prime (=maximal) clopen filter. Herrlich/Steprans [1997] and Note 10. |
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| 14 CT | Every open filter in a topological space iscontained in a prime open filter. Herrlich/Steprans [1997]and Note 10. |
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| 14 CU | The Ascoli Theorem. If \(X\) is a locally compactHausdorff space, \(Y\) is a metric space, \(C_\infty(X,Y)\) is the spaceof all continuous functions from \(X\) to \(Y\) with the compact-opentopology, and \(F\) is a subspace of \(C_\infty(X,Y)\) then the followingconditions are equivalent:\itemitem{(1)} \(F\) is compact\itemitem{(2)} (a) For each \(x\in X\), the set \(F(x) =\{f(x): f\in F\}\)is compact in \(Y\).\itemitem{}(b) \(F\) is closed in the product space \(Y^X\),\itemitem{}(c) \(F\) is equicontinuous.\par Herrlich [1997b] and Note 10. |
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| 14 CV | The Ascoli Theorem for B compactness. See form[14 CU], Note 6, and Herrlich [1997b]. |
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| 14 CW | The projective limit of a directed projective systemof non-empty, compact, \(T_2\) topological spaces is a non-empty, compact,\(T_2\) topological space. Rav [1976] and Note 71. (Comparewith [14 CD] through [14 CG].) |
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| 14 CX | The projective limit of a directed projective systemof finite topological spaces with the discrete topology is a non-empty,compact, \(T_2\) topological space. Rav [1976] and Note 71.(Compare with [14 CD] through [14 CG].) |
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| 14 CY | Let\(\)G_i(x_{i_1}, x_{i_2}, \ldots, x_{i_k}), i\in I\tag* \(\)be a system of equations in the variables \(\{x_j: j\in V\}\)with the following properties:\itemitem{(1)} For each \(j\in V\), the variable \(x_j\), has a finite domain,\(D_j\).\itemitem{(2)} Given any finite number of variables, there is an equationin the system (*) which contains those variable and possibly others.\itemitem{(3)} For each equation in the system (*) there is an equation in(*) with the same, or possibly more, variables which has a solution. \item{}Then, the system (*) has a partial solution. (The family of equations\((x_j=d_j)_{j\in V}\) with \(d_j\in D_j\) is called a partialsolution of the system (*) if for every finite subfamily of equations\(\)x_i = d_i, \ldots, x_k = d_k,\tag**\(\) there isan equation in the system (*) with the variables \(x_i, \ldots, x_k\), andpossibly other variables, such that (**) is part of the completesolution of that equation.) Abian [1973] and Note 149. |
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| 14 CZ | Wallman's Lemma. Assume \(\Cal C =\{ C_j : j\in J\}\) is a family of finite sets. For each \(j\in J\),let \(b_j =\bigcup C_j\) and assume that the family \(\{ b_j : j\in J\}\)has the finite intersection property. Then there is a choicefunction \(f\) for \(\Cal C\) such that \(\{ f(C_j) : j\in J\}\) hasthe finite intersection property. Cowen [1983]. |
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| 14 DA | Keimel's Representation Theorem. Any hyperarchimedian\(l\)-group can be imbedded into a Boolean product of a family of simple,abelean \(l\)-groups. Gluschankof [1995] and Note 151. |
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| 14 DB | If \(G\) is an \icopy{achimedian \(l\)-group} with a weakunit, then \(G\) can be imbedded into \(D(X)\) where \(X\) is a locally compacttopological space and \(D(X)\) is the set of continuous maps\(f: X\to \Bbb R \cup \{-\infty, \infty\}\) such that the inverse imageof \(\Bbb R\) is dense in \(X\). Gluschankof [1995] and Note 151. |
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| 14 DC | \icopy{Clifford's Theorem}. An \icopy{abelean \(l\)-group}is isomorphic to a subdirect product of totally ordered abelian groups.Gluschankof [1995] and Note 151. |
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| 14 DD | \icopy{Lorenzen's Theorem}. A representable \(l\)-group isisomorphic to a subdirect product of totally ordered groups.Gluschankof [1995] and Note 151. |
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| 14 DE | \icopy{Holland's Theorem}. Any \(l\)-group is isomorphicto a subdirect product of transitive \(l\)-groups. Gluschankof [1995]and Note 151. |
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| 14 DF | Any \(l\)-group can be imbedded into the \(l\)-preservingpermutations of a totally ordered set. Gluschankof [1995] andNote 151. |
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| 14 DG | In any hyperarchimedian \(l\)-group any proper\(l\)-ideal can be extended to a prime one. Gluschankof [1989] and Note 151. |
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| 14 DH | Tychonoff's Compactness Theorem for Families ofTwo Element Sets. Products of two element sets are compact.Parovi\v cenko [1969]. |
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| 14 DI | A topological space \((X,\Cal T)\) is Hausdorff if andonly if every ultrafilter in \(X\) has at most one limit. Gutierres [2004] and Note 6. |
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