Monoidal Intervals of Partial Clones

Let 2 = {0,1} and let M be a monoid on 2. The monoidal interval determined by M is the set of all partial clones on 2 whose foundation is M. We study monoidal intervals determined by several monoids on 2. Among other things, we show that if M is a monoid of total functions and contains a non-trivial unary function on 2, then the monoidal interval determined by M consists of total clones on 2. Moreover, we exhibit 4 monoids on 2 each of which determines a monoidal interval of continuum cardinality on 2.