Thymic central tolerance is essential to preventing autoimmunity. In medullary thymic epithelial cells (mTECs), the Autoimmune regulator (Aire) gene plays an essential role in this process by driving the expression of a diverse set of tissue-specific antigens (TSAs), which are presented and help tolerize self-reactive thymocytes. Interestingly, Aire has a highly tissue-restricted pattern of exp...

Lattices with a unique essential element are investigated. It is shown that, a compactly generated lattice with a unique essential element different from the top element, and with finite Goldie dimension has finite length. Mathematics Subject Classification: Primary 06C15, secondary 16P20

An essential element of a 3–connected matroid M is one for which neither the deletion nor the contraction is 3–connected. Tutte’s Wheels and Whirls Theorem proves that the only 3–connected matroids in which every element is essential are the wheels and whirls. In an earlier paper, the authors showed that a 3–connected matroid with at least one non-essential element has at least two such element...