Connection Between Both Definitions
An order theoretic meet-semilattice 〈S, ≤〉 gives rise to a binary operation ∧ such that 〈S, ∧〉 is an algebraic meet-semilattice. Conversely, the meet-semilattice 〈S, ∧〉 gives rise to a binary relation ≤ that partially orders S in the following way: for all elements x and y in S, x ≤ y if and only if x = x ∧ y.
The relation ≤ introduced in this way defines a partial ordering from which the binary operation ∧ may be recovered. Conversely, the order induced by the algebraically defined semilattice 〈S, ∧〉 coincides with that induced by ≤.
Hence both definitions may be used interchangeably, depending on which one is more convenient for a particular purpose. A similar conclusion holds for join-semilattices and the dual ordering ≥.
Read more about this topic: Semilattice
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