Every involution semigroup of order four is finitely based, so five is the minimal order for non-finitely based examples.
Math Sb N Ser 96(138), 543–559 (1975) [Russian; English translation: Math ematics of the USSR-Sbornik, 25, 507–523 (1975)]
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Finite basis problem for involution semigroups of order four
Every involution semigroup of order four is finitely based, so five is the minimal order for non-finitely based examples.