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Right pre-dioids and dioids.

A right-dioid is a semiring with a right-canonical pre-order relation relative to <>: a <~ b iff b ≡ a <> c for some c.

In other words we have that:

a <~ (a <> b) ≡ True

Consequently <~ is both reflexive and transitive:

a <~ a ≡ True
a <~ b && b <~ c ==> a <~ c ≡ True

Finally <~ is an order relation:

(a =~ b) == (a == b)

See Property