Description:
Definitions for forms [14 R] through [14 U],
[14 BH] through [14 BL], and
[86 B]. See Banaschewski [1981],
Banaschewski [1983], Banaschewski [1997],
Paseka [1989], and Vickers [1989].
Content:
Definitions for forms [14 R] through [14 U],
[14 BH] through [14 BL], and
[86 B]. See Banaschewski [1981],
Banaschewski [1983], Banaschewski [1997],
Paseka [1989], and Vickers [1989].
Definition:
- A local lattice (or frame or complete Heyting algebra) is a complete lattice \(L\) satisfying the distribution law:
\(x\wedge \bigvee_{i\in I} x_{i} = \bigvee_{i\in I}(x\wedge x_{i})\), for all \(x\), \(x_i\in L\) and \(\{x_i: i\in I\} \subseteq L\).
(Example: \({\cal D}X\) - the open sets of a topological space)
-
A lattice \(L\) is spatial if for some topological space\(X\), \(L\) is isomorphic to \({\cal D}X\). (Theorem. Every complete Boolean
algebra is a local lattice. A complete Boolean algebra is a spatial iff it is atomic.)
-
\(c\in L\) (\(L\) a local lattice) is compact if and only if \(c\le\bigvee_{i\in I} x_i\) implies \(c\le x_{i_{1}}\vee\ldots\vee x_{i_{n}}\),
for some indices \(i_{1}, \ldots , i_{n}\).
-
\(L\) is compact iff its top element \(e\) is compact.
-
\(L\) is coherent iff (i) Each element of \(L\) is the join of compact elements and (ii) the meet of any finite set, including
the empty set, of compact elements is compact.
-
\(L\) is regular iff for each \(x\in L\), \(x\) is the join of all \(y\ll x\), where \(y\ll x\) means \((\exists z\in L)(y\wedge z = 0\)
and \(x \vee z = e)\).
-
\(L\) is \(0\) dimensional iff for each \(x\in L\), \(x =\bigvee \{y\in L: y^* \vee y = e\) and \(y\le x\}\). (\(y^*\) is
the pseudo-complement of \(y\), \(y^*= \bigvee\{c\in L: c \land a = 0\}\).)
-
\(L\) is conjunctive iff \(\forall a,b\in L\), such that \(a
-
A closed set \(F\) of a topological space \(X\) is irreducible iff whenever \(G_i\), \(1\le i\le n\), are closed sets with
\(F\subseteq \bigcup_{i=1}^n G_i\), then \(F \subseteq G_i\) for some \(i\).
-
If \(F\) is a closed set in a topological space \(X\), then \(x\in X\) is a generic point for \(F\) iff \(F\) is the closure of\(\{x\}\).
-
A topological space is sober iff every irreducible closed set has a unique generic point.
-
The sum of two frames \(L_1\) and \(L_2\) is the lattice whose set is the Cartesian product of the two frames with operations
done componentwise.
-
A ring \(R\) is regular iff \((\forall a)(\exists b)(aba=a)\). (It is known that every prime ideal in a commutative regular ring is
maximal.)
-
A Gelfand algebra is a commutative Banach algebra \(A\) with unit 1 over the real numbers \({\Bbb R}\) such that
\(\Vert a^{2}\Vert= \Vert a\Vert^{2}\) and \(1 + a^{2}\) is invertible for all \(a \in A\).
-
A \(\aleph_\alpha\)-frame is a partially ordered set\(A\) with zero (0) and unit (e) in which every pair of elements has a meet and every
subset of cardinality \(\le \aleph_\alpha\) has a join and which satisfies the distributive law:
\(a\land \bigvee S =\bigvee \{a\land t : t\in S\}\) for any \(a\in A\) and subset \(S\) of \(A\)of cardinality \(\le \aleph_\alpha\).
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A cover of a lattice \(L\) is a subset \(S\) of \(L\) such that \(\bigvee S = e\) where \(e\) is the unit in the lattice.
A subcover of S is a subset of \(S\) which is still a cover.
-
A lattice \(L\) is \(\aleph_\alpha\)-Lindelöf if every cover of \(L\) has a subcover of cardinality
\(\le \aleph_\alpha\).
-
For any \(\aleph_\alpha\)-frame \(A\) and ideal \(J\) of \(A\),let \(k_0(A) = \{\bigvee S : S\subseteq A\land |S|\le\aleph_\alpha\}\).
Then \(\{ J : J \hbox{ is an ideal in } A\hbox{ and } k_0(J) = J \}\) ordered by \(\subseteq\) is a frame and is called
the frame envelope of \(A\).
Howard-Rubin number:
29
Type:
Definitions
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