A first-order logic for multi-algebras.
Hilbert algebras as implicative partial semilattices
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...
Relatively pseudocomplemented Hilbert algebras
We characterise those Hilbert algebras that are relatively pseudocomplemented posets.
Pseudocomplements in sum-ordered partial semirings
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...
Adjoint Semilattice and Minimal Brouwerian Extensions of a Hilbert Algebra
Let be a Hilbert algebra. The monoid of all unary operations on generated by operations , which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of . This semilattice is isomorphic to the semilattice of finitely generated filters of , it is subtractive (i.e., dually implicative), and its ideal lattice is isomorphic to the filter lattice of . Moreover, the order dual of the adjoint semilattice is a minimal Brouwerian extension of , and the...
Betweenness spaces and tree algebras
On JP-semilattices of Begum and Noor
In recent papers, S. N. Begum and A. S. A. Noor have studied join partial semilattices (JP-semilattices) defined as meet semilattices with an additional partial operation (join) satisfying certain axioms. We show why their axiom system is too weak to be a satisfactory basis for the authors' constructions and proofs, and suggest an additional axiom for these algebras. We also briefly compare axioms of JP-semilattices with those of nearlattices, another kind of meet semilattices with a partial join...
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