Left-invariant measures on topological $n$-ary subsemigroup of binary groups

Convolutions of measures and functions, as well as the Fourier transform of measures on locally compact Abelian $n$-ary groups were introduced in [1]. Development of harmonic analysis on $n$-ary algebraic objects endowed with a topology is closely related to the existence of a non-zero invariant measure on such objects. Invariant measures on topological $n$-ary semigroups were considered in [2] and [3].
In Theorem 2 of this paper, we establish necessary and sufficient conditions for the existence of a left-invariant measure on topological $n$-ary subsemigroups of binary groups. It can be treated as an extension of the results of [4] to the case of $n$-ary topological semigroups. The result established in Theorem 1 establishes is interesting for topological algebra.