There are three relationships presented in "My Last Duchess." One is the implied relationship between men of the aristocracy. The Duke speaks to an emissary of a nobleman whose daughter he intends to marry. From the Duke's monologue, the reader gets the impression that at least this one negotiation of a marriage has more to do with money and marital conditions than it does with love.
The second relationship is the future relationship between the Duke and his wife-to-be. It would seem to be much like the one the Duke had with his former wife, a relationship in which she must be submissive to the Duke.
The main relationship described in this poem is between the Duke and his late wife. He describes her as an outgoing woman but his jealousy paints a picture of her as inappropriately flirtatious or perhaps even as an unfaithful wife.
A heart - how shall I say? -too soon made glad,
Too easily impressed; she liked whate'er
She looked on, and her looks went everywhere. (21-24)
The Duke's jealous assessment of his late wife and his description of her as an object reveal that he thinks of her as a possession. There is even the possibility that when she did not submit to his commands, he had something to do with her death.
Whene'er I passed her; but who passed without
Much the same smile? This grew; I gave commands;
Then all smiles stopped together. (44-46)
Finally, once again discussing the portrait of his late wife, an art object, he turns the attention of the emissary to a sculpture of Neptune taming a sea horse. The juxtaposition of this piece and his wife's portrait speak to his way of thinking that women are like objects or at best, as people, something to be tamed.