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The meet formula for pseudocomplement and an aspect of the coframe of sublocales

Description:

A frame is a complete Heyting algebra; in particular it is pseudocomplemented and we have the formula

a^* = \bigvee{x : x \wedge a = 0}.

Dually, in a coframe we have the formula for the supplement

a^# = \bigwedge{x : x v a = 1}. (#)

Somewhat surprisingly, under suitable (and very weak) conditions the formula (#) holds for the pseudocomplement in a frame as well, and more generally (again under suitable conditions) we can borrow the formula for the difference (coHeyting operation)

b \ a = \bigvwedge{x : x v a = 1; x \ge b}

for the Heyting operation a->b. The conditions under which this holds true will be analyzed. As a corollary, one has the theorem on complementarity of the linear elements.
The complete lattice S(L) of sublocales of a frame is a coframe. A frame is said to be scattered if S(L) is a frame (in view of the formula above this happens iff it is a Boolean algebra). An aspect of scatteredness concerning general meet-representation of sublocales will be discussed.

Date:  2015-09-09
Start Time:   14:30
Speaker:  Aleš Pultr (Charles Univ., Prague, Czech Republic)
Institution:  Charles Univ., Prague, Czech Republic
Place:  Room 5.5
Research Groups: -Algebra, Logic and Topology
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