λῆξις · lēxis — LSJ
determination or appointment by lot, ἀρχῆς Pl. Lg. 765d.
portion assigned by lot, allotment, ib. 740a, 747e, Criti. 109c, 113b; cf. λάξις.
section of a body, determined by lot, Arist. Ath. 30.3, 31.3.
pl., fortunes, αἱ διάφοροι τῆς ἀνθρωπείας ψυχῆς λήξεις Simp. in de An. 4.6.
without the notion of lot or chance, assigned sphere, θεῶν καὶ τῶν εἰς θείαν λῆξιν πορευθέντων Jul. Or. 6.187c; ἡ ἑτέρα λ. the other world, Hld. 2.24; ἡ ἑῴα λ. the eastern part of the empire, Procop.Gaz. Pan. 497.3; ἡ τῆς Ἀθηνᾶς λ. the sphere of Athena, Lib. Or. 12.36, Decl. 1.175, Ep. 1065.3; province, τὰς λ. [τῶν θεῶν] Herm. in Phdr. p.74A.; dwelling-place, εἰ . . αἱ λ. τῶν ψυχῶν . . ὑποσέληνοί εἰσιν ib. p.104A.
as law-term, λ. δίκης or λ. alone, written complaint lodged with the Archon, as the first step in private actions, nearly, = ἔγκλημα, Pl. R. 425d, Is. 11.10, Aeschin. 1.63, cf. λαγχάνω I.3; very rarely of public actions, as in D. 39.17.
λ. τοῦ κλήρου, an application to the Archon (required of all except direct descendants) to be put in legal possession of an inheritance, τοῦ κλήρου λαχεῖν τὴν λ. ἠξίωσεν Is. 3.2, cf. Arist. Ath. 43.4.