Semigroups of transformations and endomorphisms have emerged as powerful algebraic frameworks to elucidate the underlying structures of graphs. By harnessing the principles of semigroup theory, ...

The field of model checking has evolved into a fundamental technique for verifying the correctness of software and hardware systems. At its core, model checking systematically explores the state space ...

A translation is a movement of the graph either horizontally parallel to the \(x\)-axis or vertically parallel to the \(y\)-axis. The graph of \(y = f(x)\) where \(f(x) = x^2\) is the same as the ...

\(y = x^2 + a\) represents a translation parallel to the \(y\)-axis of the graph of \(y = x^2\). If \(a\) is positive, the graph translates upwards. If \(a\) is negative, the graph translates ...

Two kinds of extensions to David Lewin's transformation theory are proposed. First, cross-type transformations transform one sort of object to another: for instance, mappings from triads to seventh ...