A relation R on a set S is antisymmetric if mutual relatedness implies equality: whenever aRb and bRa, then a = b.
Antisymmetry is what distinguishes a partial order from a preorder. A preorder allows “ties” where a ≤ b and b ≤ a without a = b (the elements are equivalent but not identical). A partial order rules this out: the only way for two elements to be mutually related is for them to be the same element.
In a category, antisymmetry corresponds to the poset being “thin” — there is at most one morphism between any two objects. If there are morphisms a → b and b → a, their composites must be identities (by uniqueness of morphisms), so a and b are isomorphic — and in a thin category, isomorphic objects are equal.
The Heyting algebra H in the semiotic universe is a partial order: its elements are semantic values with a definite ordering, and two elements that are mutually below each other are the same element. This antisymmetry ensures that the truth values of the semiotic logic are unambiguous — there are no distinct elements that occupy the same position in the lattice.