The practice or business of being an executioner; the carrying out of executions.
From executioner (one who carries out executions, from Latin executio) plus -eering (a suffix denoting a profession or activity, as in 'engineering'). It emerged as specialized terminology.
This is an extraordinarily rare word that was used in historical texts about capital punishment—it's a grim reminder that executions were once considered a formal profession with specialized practitioners, even if that's difficult to imagine today.
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