Narrow Results

A left order on a magma (e.g. semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spaces of right orders we analyze, come from knot theory and are called quandles. Our main result establishes an interesting connection between topological properties of the space of left orders on a group, and the classical algebraic result by Conrad [4] and Łoś [13] concerning the existence of left orders.

We investigate algebraic and computability-theoretic properties of orderable magmas. A magma is an algebraic structure with a single binary operation. A right order on a magma is a linear ordering of its domain, which is right-invariant with respect to the magma operation. We use tools of computability theory to investigate Turing complexity of orders on computable orderable magmas. A magma is computable if it is finite, or if its domain can be identified with the set of natural numbers and the magma operation is computable. Interesting orderable magmas that are not even associative come from knot theory.

This paper recalls my first meeting with Patrick DeHornoy and its consequences.

Homogeneous systems had their origin in the work of Professor Michihiko Kikkawa. We define a homogeneous system η on a non-empty magma G. Then, η is afterward used to define on G a multiplication μ^(a), where a is a fixed element of G. It was shown that (G, μ^(a)) is a group if and only if η(y, z) ∘ η(x, y) = η(x, z) for all elements x, y, z of G. Let us note that η(x, y) is an application of G into itself for all elements x, y in G. It is our purpose in this paper to find another equivalent condition for which (G, μ^(a)) is a group. And we have obtained η(a, μ^(a)(x, y)) = η(a, x) ∘ η(a, y) for all elements x, y in G.