<p>Rough set theory provides a formal mathematical framework for handling knowledge uncertainty in data mining. Its core components, the upper and lower approximation operators, represent fundamental concepts within the theory. Their study within lattice theory frameworks marks an important mathematical advancement. Simultaneously, overlap functions, which are non-associative binary aggregation functions contrasting with conventional t-norms, play a pivotal role in formulating fuzzy rough approximation operators. In light of these two pivotal factors, this work introduces <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((I_{O}, O)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mi>O</mi> </msub> <mo>,</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-fuzzy rough sets (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((I_{O}, O)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mi>O</mi> </msub> <mo>,</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-FRSs for short), which are based on overlap functions over complete lattices. More specifically, we first give the definition of two approximation operators and study some essential properties. Then, we consider the connection between <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\((I_{O}, O)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mi>O</mi> </msub> <mo>,</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-FRSs and <i>L</i>-fuzzy relations. Finally, we characterize the topological characteristics of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\((I_{O}, O)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mi>O</mi> </msub> <mo>,</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-FRSs and highlight that, under some conditions, some established rough set models can be taken to be special instances of the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\((I_{O}, O)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mi>O</mi> </msub> <mo>,</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-FRSs.</p>