Existence of Non-Empty Finite Infima in Meet Semilattice

Theorem

Let $\struct {S, \preceq}$ be a meet semilattice.

Let $A$ be a non-empty finite subset of $S$.

Then $A$ admits a infimum in $\struct {S, \preceq}$.

Proof

This follows by mutatis mutandis of the proof of Existence of Non-Empty Finite Suprema in Join Semilattice.

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article YELLOW_0:55