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The use of stream based applications is in expansion in many contexts and easy and efficient data stream management is crucial for such applications. That is why numerous solutions for stream query processing have been proposed by the scientific community. Several query processors exist and offer heterogeneous querying capabilities. This paper reports a formal work on the operators behind such query processing solutions. It points out the semantic heterogeneity of some important operators and how this leads to some kind of semantic ambiguity which may affect the application semantics. This paper revisits the definition of the main operators used for stream query processing and proposes definitions which are semantically unambiguous. The main issue is the positional order of data items in a stream and its propagation across the operators. The proposed formalization deepens the understanding of stream queries and facilitates the comparison of the semantics implemented by existing systems. This paper also presents the prototype implementing our formal proposal.