Boolean Posets, Posets under Inclusion and Products of Relational Structures 1

Summary. In the paper some notions useful in formalization of[11] are introduced, e.g. the definition of the poset of subsets of a setwith inclusion as an ordering relation. Using the theory of many sortedsets authors formulate the definition of product of relational structures.MML Identifier: YELLOW 1. The terminology and notation used in this paper are introduced in the followingarticles: [19], [21], [9], [22], [24], [23], [16], [6], [7], [5], [10], [4], [13], [20], [25],[12], [2], [17], [15], [18], [3], [14], [1], and [8].1. Boolean Posets and Posets under InclusionIn this paper X will be a set.Let L be a lattice. Observe that Poset(L) has l.u.b.’s and g.l.b.’s.Let L be an upper-bounded lattice. Note that Poset(L) is upper-bounded.Let L be a lower-bounded lattice. One can check that Poset(L) is lower-bounded.Let L be a complete lattice. One can verify that Poset(L) is complete.Let X be a set. Then ⊆X is an order in X.Let X be a set. The functor hX,⊆i yielding a strict relational structure isdefined as follows:(Def. 1) hX,⊆i = hX,