Fig. 3

Road-junction Galois lattice associated to the colourized notional urban street network introduced in Fig. 1 with the labelling chosen in Fig. 2. This Galois lattice is obtained by applying the Formal Concept Analysis (FCA) paradigm to the incidence relation I whose chart representation is given in Table 1. This construction is one-to-one. A Galois lattice is an algebraic structure that underlies a partial order relation \(\preccurlyeq \) and two algebraic operators, a join operator ∨ and a meet operator ∧. The partial order relation can be interpreted as an extended logical imply relation →. The arrows in the diagram inherit this interpretation. For FCA lattices, each element is a pair of sets [R,J] where R is a set of objects and J a set of attributes. Here the roads r_∗ are the objects whose attributes are junctions j_∗ and impasses i_∗ (see Table 1). Because the roads r_∗ do not cross to each others more than once, the Galois lattice takes an intuitive two-layer form. Indeed, the join-irreducible elements [{r_∗},J] and the meet-irreducible elements [R,{j_∗}] readily identify themselves with their road r_∗ and their junctions j_∗, respectively. So, the roads r_∗ and the junctions j_∗ immediately form, respectively, the lower and upper nontrivial layers of the Galois lattice. This also gives meaningful and intuitive interpretations to the partial order relation \(\preccurlyeq \) and to the operators ∨ and ∧: \({r_{a}}\preccurlyeq {j_{7}}\) (or r_a→j₇) reads “road r_a passes through junction j₇” or “junction j₇ is along road r_a”; r_a∨r_b=j₃ reads “roads r_a and r_b join at junction j₃”; j₃∧j₇=r_a reads “junctions j₃ and j₇ meet road r_a”. Each colourized arrow in the diagram bears the colour of its road. The top element ⊤ is the urban street network as a whole, while the bottom element ⊥ is its absurd counterpart, emptiness or the absence of urban street network