Dependencies

For any \(g \in G\) and \(p \in \mathbb {N}\), there exists an integer \(q \in \mathbb {Z}\) such that

An anomalous pair in a left ordered semigroup \(S\) is a pair of elements \(a,b \in S\) such that for all positive \(n \in \mathbb {N}\), either

or

.

A left ordered semigroup is Archimedean if any two of its elements of the same sign are mutually Archimedean.

Let \(a\) and \(b\) be elements of a left ordered semigroup \(S\) that are not one.

Then \(a\) is said to be Archimedean with respect to \(b\) if there exists an \(N\in \mathbb {N}^+\) such that for \(n {\gt} N\), the inequality \(b {\lt} a^n\) holds if \(b\) is positive, and the inequality \(b {\gt} a^n\) holds if \(b\) is negative.

An Archimedean group is a left ordered group such that for any \(g, h \in G\) where \(g \ne 1\), there exists an integer \(z\) such that \(h {\lt} g^z\).

A left ordered semigroup \(S\) has large differences if for all \(a,b\in S\), the two following implications hold

• if \(a\) is positive and \(a{\lt}b\), then there exists \(c\in S\) and \(n\in \mathbb {N}^+\) such that \(c\) is Archimedean with respect to \(a\) and \(a^n*c \le b^n\)

• if \(a\) is negative and \(b {\lt} a\), then there exists \(c\in S\) and \(n\in \mathbb {N}^+\) such that \(c\) is Archimedean with respect to \(a\) and \(a^n*c \ge b^n\)

A left linear ordered group \(G\) is a left ordered group that is also a linear order.

A left ordered group \(G\) is a group and a partial order such that for all \(x, y, z \in G\), if \(x \le y\), then \(z * x \le z * y\).

A left ordered semigroup \(S\) is a semigroup and a partial order such that for all \(x, y, z\in S\), if \(x \le y\), then \(z * x \le z * y\).

A linear ordered cancel semigroup is an ordered cancel semigroup where its partial order is a linear order.

A linear ordered semigroup is an ordered semigroup where its partial order is a linear order.

An element \(a\) of a left ordered semigroup \(S\) is negative if for all \(x\in S\), \(a*x {\lt} x\).

An element \(a\) of a left ordered semigroup \(S\) is one if for all \(x\in S\), \(a*x = x\).

An ordered cancel semigroup \(S\) is an ordered semigroup such that for all \(a,b,c\in S\), if \(a * b \le a * c\) then \(b \le c\) and if \(b * a \le c * a\) then \(b \le c\).

An ordered semigroup \(S\) is a left and right ordered semigroup.

We define a map \(\phi \colon G \to \mathbb {R}\) by mapping \(g\) to the real number that \(\frac{q_g(n)}{n}\) converges to as we know from Theorem 8.

An element \(a\) of a left ordered semigroup \(S\) is positive if for all \(x\in S\), \(a*x {\gt} x\).

We define a function \(q \colon G \to \mathbb {N} \to \mathbb {R}\) using Theorem 4 such that for any \(g\in G\) and \(n \in \mathbb {N}\),

A right ordered semigroup \(S\) is a semigroup and a partial order such that for all \(x, y, z\in S\), if \(x \le y\), then \(x * z \le y * z\).

We have that \(\phi (f) = 1\) where \(f\) is our fixed positive element.

Let \(S\) be a linear ordered cancel semigroup without anomalous pairs. Then \(S\) is isomorphic to a subsemigroup of the real numbers.

If \(G\) is a left linear ordered group that is Archimedean, then \(G\) is isomorphic to a subgroup of \(\mathbb {R}\).

The map \(\phi \) is injective.

A linear ordered cancel semigroup without anomalous pairs is an Archimedean commutative semigroup.

If \(S\) is a non-Archimedean linear ordered cancel semigroup, then there exists an anomalous pair.

Let \(M\) be a linear ordered cancel commutative monoid without anomalous pairs and let \(G\) be its Grothendieck group. If \(M\) has a positive (negative) element, then for any element \(\frac{a}{b} \in G\) where \(a,b \in M\), there exist \(a', b' \in M\) such that \(a'\) and \(b'\) are positive (negative) and \(\frac{a}{b} = \frac{a'}{b'}\).

Let \(M\) be a linear ordered cancel commutative monoid. If \(M\) does not have anomalous pairs, then its Grothendieck group is Archimedean.

In a linear ordered cancel semigroup \(S\), there are no anomalous pairs if and only if there are large differences.

For all \(g,h \in G\), if \(g \le h\) then \(\phi (g) \le \phi (h)\).

The map \(\phi \) is a homomorphism.

For all \(g_1,g_2 \in G\) and \(p\in \mathbb {N}\),

Let \(S\) be a linear ordered cancel semigroup without anomalous pairs such that there exists a positive element of \(S\). Then for all \(x \in S\), there exists a \(y \in S\) such that \(y\) is positive and \(x*y\) is positive.

Similarly, if there exists a negative element of \(S\) then for all \(x \in S\) there exists a \(y \in S\) such that \(y\) is negative and \(x*y\) is negative.

For any \(g \in G\), the sequence \(\frac{q_g(n)}{n}\) converges.

For any \(g \in G\) and \(a,b \in \mathbb {N}\), we have that

For any sequence \(a_n\) of real numbers, if there exists \(C \in \mathbb {R}\) such that for all \(m,n\in \mathbb {N}\) we have that

then sequence \(\frac{a_n}{n}\) converges.

For all \(g,h \in G\), we have that \(g \le h\) if and only if \(\phi (g) \le \phi (h)\).

Each element \(a\) of a linear ordered cancel semigroup \(S\) is either positive, negative, or one.

If \(S\) is a linear ordered cancel semigroup that does not have anomalous pairs, then there exists a linear ordered commutative group \(G\) that is Archimedean such that \(S\) is isomorphic to a subsemigroup of \(G\).

If \(S\) is a linear ordered cancel semigroup without anomalous pairs, then there exists a linear ordered cancel commutative monoid \(M\) without anomalous pairs such that \(S\) is isomorphic to a subsemigroup of \(M\).