Demonic composition

Suppose ${\displaystyle R}$ is a binary relation between ${\displaystyle X}$ and ${\displaystyle Y}$ and ${\displaystyle S}$ is a relation between ${\displaystyle Y}$ and ${\displaystyle Z.}$ Their right demonic composition ${\displaystyle R{\textbf {;}}^{\to }S}$ is a relation between ${\displaystyle X}$ and ${\displaystyle Z.}$ Its graph is defined as

$${\displaystyle \{(x,z)\$$ :\ x\mathrel {(S\circ R)} z{\text{ and for all }}y\in Y\ (x\mathrel {R} y{\text{ implies }}y\mathrel {S} z)\}.}

Conversely, their left demonic composition ${\displaystyle R{\textbf {;}}^{\leftarrow }S}$ is defined by

$${\displaystyle \{(x,z)\$$ :\ x\mathrel {(S\circ R)} z{\text{ and for all }}y\in Y\ (y\mathrel {S} z{\text{ implies }}x\mathrel {R} y)\}.}

• Backhouse, Roland; van der Woude, Jaap (1993), "Demonic operators and monotype factors", Mathematical Structures in Computer Science, 3 (4): 417–433, CiteSeerX 10.1.1.40.9602, doi:10.1017/S096012950000030X, MR 1249420.