Formal Concept Analysis (FCA) is a field of applied mathematics based on formalization of the notion of concept from cognitive psychology and has been widely studied in the last several decades. From a description of objects by their features FCA derives a hierarchy of concepts which is formalized by a complete lattice called a concept lattice. We explore some fundamental aspects of FCA. First, we focus on incremental concept lattice construction and analysis of its basic step, removal of an incidence, and propose two algorithms for incremental concept lattice construction. Second, we study generated complete sublattices and show how their corresponding closed subrelations can be efficiently computed. Lastly, we investigate a new type of subrelations from which a new formal rectangle type arises, we provide motivation from cognitive psychology for it and propose a basic theorem for lattices of such rectangles.Formal Concept Analysis (FCA) is a field of applied mathematics based on formalization of the notion of concept from cognitive psychology and has been widely studied in the last several decades. From a description of objects by their features FCA derives a hierarchy of concepts which is formalized by a complete lattice called a concept lattice. We explore some fundamental aspects of FCA. First, we focus on incremental concept lattice construction and analysis of its basic step, removal of an incidence, and propose two algorithms for incremental concept lattice construction. Second, we study generated complete sublattices and show how their corresponding closed subrelations can be efficiently computed. Lastly, we investigate a new type of subrelations from which a new formal rectangle type arises, we provide motivation from cognitive psychology for it and propose a basic theorem for lattices of such rectangles.