FAIR is a non-profit organization dedicated to providing well-documented answers to criticisms of the doctrine, practice, and history of The Church of Jesus Christ of Latter-day Saints.
Question: What can you tell me about Mary Elizabeth Rollins Lightner and polyandry?
Mary Elizabeth Rollins Lightner as a polyandrous wife of Joseph Smith
Summary: Mary was sealed to Joseph while still married and living with her husband Adam Lightner. This type of sealing has been called "polyandrous."
Mary Elizabeth Rollins Lightner very unlikely to have had children fathered by Joseph
Summary: Even by the turn of the century, the LDS Church had no solid evidence of children by Joseph. "I knew he had three children," said Mary Elizabeth Lightner, "They told me. I think two of them are living today but they are not known as his children as they go by other names."[54] Again, evidence for children is frustratingly vague—Lightner had only heard rumours, and could not provide any details. It would seem to me, however, that this remark of Lightner's rules out her children as possible offspring of Joseph. Her audience was clearly interested in Joseph having children, and she was happy to assert that such children existed. If her own children qualified, why did she not mention them?
Joseph Smith's Polygamy: "Mary Elizabeth Rollins", by Brian C. Hales
Joseph Smith's Polygamy: "After Louisa Beaman, Joseph Smith Seeks Almost Exclusively “Eternity Only” Sealings", by Brian C. Hales