Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A strict total order which is not a connected relation.
A strict weak order is a strict preorder which is also a cotransitive relation.
These are sometimes called strict partial orders in the literature, but the term “strict partial order” is also used for other order relations in mathematics.
Strict weak orders are used in models of the real numbers which have infinitesimals, such as in the smooth real numbers in synthetic differential geometry, and thus where not every real number not greater than or less than zero is equal to zero.
Strict weak orders are asymmetric relations.
Transitivity of $\lt$ says that for all $x \in S$ and $y \in S$, $x \lt y$ and $y \lt x$ implies $x \lt x$. However, irreflexivity says that for all $x$, $x \leq x$ is always false. This implies that for all $x$ and $y$, $x \lt y$ and $y \lt x$ is always false, which is precisely the condition of asymmetry.
Connected strict weak orders are pseudo-orders.
The proof is the same as above, except both relevant orders are connected relations.
An important part of a strict weak order is that it is a preorder.
The negation of a strict weak order is transitive.
The contrapositive of cotransitivity says that
for all $x$, $y$, and $z$, if $x \lt y$ or $y \lt z$ is false, then $x \lt z$ is false.
By one of de Morgan's laws, that $x \lt y$ or $y \lt z$ is false is logically to equivalent to that $\neg(x \lt y)$ and $\neg(y \lt z)$, and substituting this into the original statement results in
if $\neg(x \lt y)$ and $\neg(y \lt z)$, then $\neg(x \lt z)$
which is precisely transitivity for the negation of the strict weak order.
The negation of a strict weak order is reflexive.
Irreflexivity states that $\neg (x \lt x)$ is true, which is precisely reflexivity for the negation of the strict weak order.
The negation of a strict weak order is a preorder.
The incomparability relation of a strict weak order, $\neg (x \lt y) \wedge \neg (y \lt x)$, is an equivalence relation
For every preorder, $(x \leq y) \wedge (y \leq x)$ is an equivalence relation. Since $\neg (x \lt y)$ is a preorder, $\neg (x \lt y) \wedge \neg (y \lt x)$ is an equivalence relation.
Strict weak orders are called strict partial orders in section 4.1 of:
Strict weak orders are used in defining the smooth real numbers in:
Last revised on December 26, 2023 at 05:12:19. See the history of this page for a list of all contributions to it.