Statement:
Sikorski's Extension Theorem: Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). Sikorski [1964], p. 141.
Howard_Rubin_Number: 50
Parameter(s): This form does not depend on parameters
This form's transferability is: Unknown
This form's negation transferability is: Negation Transferable
Article Citations:
Sikorski-1950: Cartesian products of Boolean algebras
Book references
Boolean Algrbras, Sikorski, R., 1960
Boolean Algrbras, 2nd ed., Sikorski, R., 1964
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 50 A | Every complete Boolean algebra is injective.Banaschewski [1988]. |
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| 50 B | Monteiro's Theorem: Let \(K\) be a complete Booleanalgebra, \(A\) a subalgebra of the Boolean algebra \(B\) and \(f\) ahomomorphism 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 \(x\) and \(y\) in\(B\), \(d(x\lor y)=d(x)\lor d(y)\)) with \(f(x)\le d(x)\) for all \(x\in A\).Then there 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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| 50 C | Every DeMorgan compact regular frame is injective inthe category of all such frames. Banaschewski [1988], notes 29and 60. |
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| 50 D | For any extension \(M \supseteq L\) of a compact regularframe \(L\), there is an \(s\in M\) maximal with respect to the property that\(x\lor s =1 \to x=1\) for all \(x\in L\). Banaschewski [1988],notes 29 and 60. |
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| 50 E | Form 14 +Form 297. Banaschewski [1988] andBell [1988]. |
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| 50 F | For any complete Boolean algebra \(B\) and any subalgebra\(A\) of \(B\), there is a \(U \in V^{(B)}\) such that \(V^{(B)}\models\)``\(U\) is an ultrafilter on \(A\) and \(U^B \subseteq U\)''. (\(V^{(B)}\) isthe Boolean valued universe constructed from \(B\), \(U^B = \{ (\hat x,x)\, : x\in B\,\}\) is the canonical ultrafilter on \(B\) and \(x\mapsto\hat x\)is the canonical embedding of \(V\) into \(V^{(B)}\).) Bell [1977]and \cite{1983}. |
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| 50 G | For any complete Boolean algebra \(B\) andany \(C\in V^{(B)}\) such that \(V^{(B)}\models\) ``\(C\) is a Boolean algebra.''there is a \(U\in V^{(B)}\) such that \(V^{(B)}\models \) ``\(U\) is anultrafilter in \(C\).'' See [50 F] for definitions. Bell [1983]. |
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| 50 H | If \(m: A\to B\) is a monomorphism between Booleanalgebras, then there is an \(m\)-minimal epimorphism \(p: B\to C\) to someBoolean algebra \(C\). (\(p\) is \(m\)-minimal if (1) \(p\circ m\) is monic and(2) if \(q:C\to D\) is any epimorphism to a Boolean algebra \(D\) such that\(q\circ p\circ m\) is monic, then \(q\) is an isomorphism.) Bell [1988]. |
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| 50 I | If \(A\) is a subalgebra of a Boolean algebra \(B\), thenthere is a \(\subseteq\)-maximal proper filter \(F\) in \(B\) such that \(F\cap A=\{ 1\}\). Bell [1988] |
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| 50 J | Form 299 +Form 14. Bell [1988]. |
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| 50 K | Form 300 +Form 14. Bell [1988]. |
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| 50 L | Form 301 +Form 14. Bell [1988]. |
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| 50 M | Form 302 +Form 14. Bell [1988]. |
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