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Hilbert algebras as implicative partial semilattices

100%
Cīrulis J.
Open Mathematics
|
2007
|
tom 5
|
nr 2
264-279
EN
The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra i ?both algebras have the same ?lters.An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice),and conversely.The implication in an implicative partial semilattice is characterised in terms of ?lters of the underlying partial semilattice.
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Pseudocomplements in sum-ordered partial semirings

100%
Cīrulis J.
Discussiones Mathematicae - General Algebra and Applications
|
2007
|
tom 27
|
nr 2
169-186
EN
We study a particular way of introducing pseudocomplementation in ordered semigroups with zero, and characterise the class of those pseudocomplemented semigroups, termed g-semigroups here, that admit a Glivenko type theorem (the pseudocomplements form a Boolean algebra). Some further results are obtained for g-semirings - those sum-ordered partially additive semirings whose multiplicative part is a g-semigroup. In particular, we introduce the notion of a partial Stone semiring and show that several well-known elementary characteristics of Stone algebras have analogues for such semirings.
3
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Relatively pseudocomplemented Hilbert algebras

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Cīrulis J.
Open Mathematics
|
2008
|
tom 6
|
nr 1
189-190
EN
We characterise those Hilbert algebras that are relatively pseudocomplemented posets.
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