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(set theory) A partial order.

Fig. 1 The Hasse diagram of the set of all subsets of a three-element set ${\displaystyle \{x,y,z\},}$ ordered by inclusion. Sets connected by an upward path, like ${\displaystyle \emptyset }$ and ${\displaystyle \{x,y\}}$, are comparable, while e.g. ${\displaystyle \{x\}}$ and ${\displaystyle \{y\}}$ are not.

(set theory) A partial order.

Fig. 2 Commutative diagram about the connections between strict/non-strict relations and their duals, via the operations of reflexive closure (cls), irreflexive kernel (ker), and converse relation (cnv). Each relation is depicted by its logical matrix for the poset whose Hasse diagram is depicted in the center. For example ${\displaystyle 3\not \leq 4}$ so row 3, column 4 of the bottom left matrix is empty.

(set theory) A partial order.

Fig. 3 Graph of the divisibility of numbers from 1 to 4. This set is partially, but not totally, ordered because there is a relationship from 1 to every other number, but there is no relationship from 2 to 3 or 3 to 4