The author considers the sets \(C(\alpha) = \{z \in \mathbb{C} : | z \sin\;\alpha \pm i \cos\;\alpha | \leq 1\}\) and proves that these sets with \(\alpha \in (0, \pi/2)\) form multiplicative semigroups in the complex plane. The main result of the paper stands that the semigroups \(C(\alpha)\) and \(C(\beta)\) are not isomorphic for \(\alpha \not= \beta\) and if \(\Phi\) is an automorphism of \(C(\alpha)\), then either \(\Phi(z) = z\) for all \(z \in C(\alpha)\) or \(\Phi(z) = {\bar z}\) for all \(z \in C(\alpha)\). The author also presents a detailed study of all continuous semicharacters of the semigroups \(C(\alpha)\) and of all continuous automorphisms of the closed unit disk.