Let \(S\) be a regular \(*\)-semigroup, that is, a semigroup with involution \(a\mapsto a^{-1}\) for which \(a^{-1}\) is an inverse of \(a\). Let \(E_S\) denote the set of idempotents of \(S\) and \(P_S\) the set of \textit{projections} of \(S\), that is, \(P_S=\{e\in E_S:e=e^{-1}\}\). For any regular \(*\)-semigroup consider the induced unary semigroups \((S,\cdot,{^+})\), \((S,\cdot,{^*})\) and bi-unary semigroup \((S,\cdot,{^+},{^*})\), where \(a^+=aa^{-1}\) and \(a^*=a^{-1}a\). Then \(P_S=\{a^+:a\in S\}=\{a^*:a\in S\}\). For any regular \(*\)-semigroup the unary semigroup \((S,\cdot,{^*})\) satisfies: (1) \(xx^*=x\); (2) \((xy)^*=(x^*y)^*\); (3) \((x^*y^*)^*=y^*x^*y^*\); (4) \(x^*x^*=x^*\). The unary semigroup \((S,\cdot,{^+})\) satisfies the dual identities \((1^r)\)-\((4^r)\) (with \(^+\) substituted for \(^*\)). The bi-unary semigroup \((S,\cdot,{^+},{^*})\) satisfies the identities (1) through (4), \((1^r)\) through \((4^r)\) and, in addition: (5) \((x^+)^*=x^+\) and \((x^*)^+=x^*\); (6) \((xy)^+x=xy^+x^*\); and \(x(yx)^*=x^+y^*x\). Any semigroup \((S,\cdot,{^*})\) that satisfies the identities (1)-(4) is called a \textit{right \(P\)-Ehresmann semigroup}. The set \(P_S=\{a^*:a\in S\}\) is the set of \textit{projections} of \(S\). A \textit{left \(P\)-Ehresmann semigroup} is a semigroup \((S,\cdot,{^+})\) that satisfies the identities \((1^r)\)-\((4^r)\), in which the set of projections is \(P_S=\{a^+:a\in S\}\). A \(P\)-\textit{Ehresmann semigroup} is a semigroup \((S,\cdot ,{^+},{^*})\) that is a left \(P\)-Ehresmann semigroup under \(^+\), a right \(P\)-Ehresmann semigroup under \(^*\), and in addition satisfies the identities (5). A \(P\)-\textit{restriction semigroup} is a \(P\)-Ehresmann semigroup that, in addition, satisfies the `generalized ample' identities (6). For a given right \(P\)-Ehresmann semigroup \(S\), an operation \(\star\) is induced on the poset \(P_S\) by the rule \(e\star f=fef\), thereby defining `right projection algebras'. The operation \(\star\) will not in general be associative. These algebras are characterized by axioms. With any right projection algebra \(P\) one can associate a right \(P\)-Ehresmann semigroup that is a `large' subsemigroup \(\text{Ord}_1P\) of the semigroup of order-preserving transformations of \(P\). In the case of left \(P\)-Ehresmann semigroups, the operation \(\times\) is defined dually. Then any right \(P\)-Ehresmann semigroup \(S\) is represented in the semigroup \(\text{Ord}_1P_S\), in such a way that an algebra-isomorphism is induced between the respective right projection algebras. In the two-sided case, for the `projection algebras' \((P_S,\times,\star)\), the operations are the reverses of each other. This entails the construction from any projection algebra \(P\) of a `Munn-type' semigroup \(T_P\), consisting of the algebra isomorphisms between principal ideals of \(P\). This construction generalizes the Munn semigroup of a semilattice. The resulting semigroup is in fact a regular \(*\)-semigroup. Any \(P\)-restriction semigroup \(S\) is represented as a full subsemigroup of \(T_{P_S}\), in such a way that the projection algebra of \(T_{P_S}\) is algebra-isomorphic to \(P_S\). By further characterizing the sets of projections `internally', the author connects his universal algebraic approach with the classical approach of the so-called `York school' [see \textit{C. Hollings}, Eur. J. Pure Appl. Math. 2, No. 1, 21-57 (2009; Zbl 1214.20056)].