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In this paper, we develop the theory of abstract crystals for quantum Borcherds–Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of    B  (  ∞  )    and    B  (  λ  )    as its application, where    B  (  ∞  )    and    B  (  λ  )    are the crystals of the negative half part of the quantum Borcherds–Bozec algebraU  q    (  g  )and its irreducible highest weight module    V  (  λ  )   , respectively.

Abstract crystals for quantum Borcherds–Bozec algebras