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The (co)associative, partially (co)associative and totally (co)associative$3$-ary (co) algebras and infinitesimal bialgebras are constructed anddiscussed. Their trimodules and matched pairs are defined and completelycharacterized. The main structural properties and relations are also deducedand analyzed.

(Co)associative $3$-ary (co)algebras and infinitesimal bialgebras: construction and main properties