There are 18 inequivalent ways to define an associative operation on three elements: while there are, altogether, a total of 39 = 19683 different binary operations that can be defined, only 113 of these are associative, and many of these are isomorphic or antiisomorphic so that there are essentially only 18 possibilities.
One of these non-commutative bands results from adjoining an identity element to LO2, the left zero semigroup with two elements (or, dually, to RO2, the right zero semigroup).
It is sometimes called the flip-flop monoid, referring to flip-flop circuits used in electronics: the three elements can be described as "set", "reset", and "do nothing".
[3] The irreducible elements in this decomposition are the finite simple groups plus this three-element semigroup, and its subsemigroups.
(The equation x4 = x describes C3, the group with three elements, already mentioned.)
There are also two other anti-isomorphic pairs of non-commutative non-band semigroups.
Monogenic semigroup (index 2, period 2) Subsemigroup: {y,z} ≈ C2 3.
Aperiodic monogenic semigroup (index 3) Subsemigroup: {y,z} ≈ O2 4.
idempotent semigroup (left flip-flop monoid) Subsemigroups: {x,y} ≈ LO2, {x,z} ≈ {y,z} ≈ CH2 18B.
its opposite (right flip-flop monoid) Index of two element subsemigroups: C2: cyclic group, O2: null semigroup, CH2: semilattice (chain), LO2/RO2: left/right zero semigroup.