| Summary: | Let <i>S</i> be a Clifford semigroup and <i>A</i> a subset of <i>S</i>. We write <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>a</mi><mi>y</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> for the Cayley digraph of a Clifford semigroup <i>S</i> relative to <i>A</i>. The (weak, path, weak path) independence number of a graph is the maximum cardinality of an (weakly, path, weakly path) independent set of vertices in the graph. In this paper, we characterize maximal connected subdigraphs of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>a</mi><mi>y</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and apply these results to determine the (weak, path, weak path) independence number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mi>a</mi><mi>y</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>.
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