| Title:
|
Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra (English) |
| Author:
|
Cīrulis, Jānis |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
51 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
41-51 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $A := (A,\rightarrow ,1)$ be a Hilbert algebra. The monoid of all unary operations on $A$ generated by operations $\alpha _p\colon x \mapsto (p \rightarrow x)$, which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of $A$. This semilattice is isomorphic to the semilattice of finitely generated filters of $A$, it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of $A$. Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of $A$, and the embedding of $A$ into this extension preserves all existing joins and certain “compatible” meets. (English) |
| Keyword:
|
adjoint semilattice |
| Keyword:
|
Brouwerian extension |
| Keyword:
|
closure endomorphism |
| Keyword:
|
compatible meet |
| Keyword:
|
filter |
| Keyword:
|
Hilbert algebra |
| Keyword:
|
implicative semilattice |
| Keyword:
|
subtraction |
| MSC:
|
03G25 |
| MSC:
|
06A12 |
| MSC:
|
06A15 |
| MSC:
|
08A35 |
| idZBL:
|
Zbl 06204929 |
| idMR:
|
MR3058872 |
| . |
| Date available:
|
2012-11-26T10:16:42Z |
| Last updated:
|
2014-03-12 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143066 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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