Specifically with regard to Mawgarn’s Axe which permits defeating a monster when transferring by way of a dungeon space.

Unfortunately Defeat is just not addressed inside the Lexicon.

Mawgarn's Axe - Choose a monster, for each level of the wielding hero, roll 1d6+1. If you roll lower than the Mosnter's HP, destroy this Hero. Otherwise, defeat it the next time this turn you move into of through its room.