Skew Boolean Algebras

Maximal normal bands in rings form noncommutative variants of (generalized) Boolean algebras under multiplication and the cubic join $x\bigtriangledown y=x+y+yx-xyx-yxy$. These were called by J. Leech Skew Boolean Algebras but their studied started already in 1980 with W. Cornish generalizing relatively complemented distributive lattices. In this talk we are going to report on the work of this two authors together with M. Spinks, R. Veroff and R. Bignall, in the variety of skew Boolean algebras with finite intersections and their important connections with other structures such as BCK algebras, discriminator varieties and Boolean products.