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Mathematical Aspects of Logic Programming Semantics
order induced on A by the operation +. It is immediate that A equipped
with this partial order is an ordered semigroup, as just defined.)
(2) For each a ∈ A, there is a unique b (=
a
A
b b a
2 ) ∈ such that + = .
(3) For all a, b ∈ A, the infimum a ∧ b of a and b exists in A relative to the
partial order ≤ defined in (1).
(4) For all a, b, c ∈ A, (a ∧ b) + c = (a + c) ∧ (b + c).
Note that if {(A i , + i , 0 i , ∞ i ) | i ∈ I} is a family of value semigroups, then
so is their product (A, +, 0, ∞), where +, 0, and ∞ are defined coordinatewise.
4.1.3 Definition A set P of positives in a value semigroup A is a subset P
of A satisfying the following axioms.
(1) If r, s ∈ P , then r ∧ s ∈ P .
(2) If r ∈ P and r ≤ a, then a ∈ P .
(3) If r ∈ P , then
r
2 ∈ P .
(4) If a ≤ b + r for all r ∈ P , then a ≤ b.
4.1.4 Example The set R of extended real numbers [0, ∞] together with
addition forms a value semigroup, the set (0, ∞] is a set of positives for this
example, and the induced partial order ≤ is the usual one on R.
4.1.5 Definition A continuity space is a quadruple X = (X, d, A, P ), where
X is a non-empty set, A is a value semigroup, P is a set of positives in A,
and d : X × X → A is a function, called a continuity function, satisfying the
following axioms.
(1) For all x ∈ X, d(x, x) = 0.
(2) For all x, y, z ∈ X, d(x, z) ≤ d(x, y) + d(y, z).
Finally, we define the topology generated by a continuity space.
4.1.6 Definition Suppose that X = (X, d, A, P ) is a continuity space. Let
x ∈ X, and let b ∈ P . Then B b (x) = {y ∈ X | d(x, y) ≤ b} is called the ball
of radius b about x. The topology T (X ) generated by X consists of all those
subsets O of X satisfying the property: if x ∈ O, then B b (x) ⊆ O for some
b ∈ P .
The main result concerning continuity spaces is the following theorem due
to R. Kopperman [Kopperman, 1988].
Mathematical Aspects of Logic Programming Semantics
order induced on A by the operation +. It is immediate that A equipped
with this partial order is an ordered semigroup, as just defined.)
(2) For each a ∈ A, there is a unique b (=
a
A
b b a
2 ) ∈ such that + = .
(3) For all a, b ∈ A, the infimum a ∧ b of a and b exists in A relative to the
partial order ≤ defined in (1).
(4) For all a, b, c ∈ A, (a ∧ b) + c = (a + c) ∧ (b + c).
Note that if {(A i , + i , 0 i , ∞ i ) | i ∈ I} is a family of value semigroups, then
so is their product (A, +, 0, ∞), where +, 0, and ∞ are defined coordinatewise.
4.1.3 Definition A set P of positives in a value semigroup A is a subset P
of A satisfying the following axioms.
(1) If r, s ∈ P , then r ∧ s ∈ P .
(2) If r ∈ P and r ≤ a, then a ∈ P .
(3) If r ∈ P , then
r
2 ∈ P .
(4) If a ≤ b + r for all r ∈ P , then a ≤ b.
4.1.4 Example The set R of extended real numbers [0, ∞] together with
addition forms a value semigroup, the set (0, ∞] is a set of positives for this
example, and the induced partial order ≤ is the usual one on R.
4.1.5 Definition A continuity space is a quadruple X = (X, d, A, P ), where
X is a non-empty set, A is a value semigroup, P is a set of positives in A,
and d : X × X → A is a function, called a continuity function, satisfying the
following axioms.
(1) For all x ∈ X, d(x, x) = 0.
(2) For all x, y, z ∈ X, d(x, z) ≤ d(x, y) + d(y, z).
Finally, we define the topology generated by a continuity space.
4.1.6 Definition Suppose that X = (X, d, A, P ) is a continuity space. Let
x ∈ X, and let b ∈ P . Then B b (x) = {y ∈ X | d(x, y) ≤ b} is called the ball
of radius b about x. The topology T (X ) generated by X consists of all those
subsets O of X satisfying the property: if x ∈ O, then B b (x) ⊆ O for some
b ∈ P .
The main result concerning continuity spaces is the following theorem due
to R. Kopperman [Kopperman, 1988].