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By Stankey Burris, H. P. Sankappanavar

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The definition given above is not the one which appears in standard texts on groups, for they use only one binary operation and axioms involving existential quantifiers. The reason for the above choice, and for the descriptions given below, will become clear in §2. Groups are generalized to semigroups and monoids in one direction, and to quasigroups and loops in another direction. (2) Semigroups and Monoids. A semigroup is a groupoid G, · in which (G1) is true. It is commutative (or Abelian) if (G4) holds.

Show that B, ∨, ∧, →, 0, 1 is a Heyting algebra. 4. Show that every Boolean algebra is an ortholattice, but not conversely. 5. (a) If H, ∨, ∧, →, 0, 1 is a Heyting algebra and a, b ∈ H show that a → b is the largest element c of H (in the lattice sense) such that a ∧ c ≤ b. (b) Show that the class of bounded distributive lattices L, ∨, ∧, 0, 1 such that for each a, b ∈ L there is a largest c ∈ L with a ∧ c ≤ b is precisely the class of reducts of Heyting algebras (to {∨, ∧, 0, 1}). §2. Isomorphic Algebras, and Subalgebras 31 (c) Show how one can construct a Heyting algebra from the open subsets of a topological space.

8. An algebra A is congruence-distributive (congruence-modular) if Con A is a distributive (modular) lattice. If θ1 , θ2 ∈ Con A and θ1 ◦ θ2 = θ2 ◦ θ1 then we say θ1 and θ2 are permutable, or θ1 and θ2 permute. A is congruence-permutable if every pair of congruences on A permutes. A class K of algebras is congruence-distributive, congruence-modular, respectively congruence-permutable iff every algebra in K has the desired property. We have already looked at distributivity and modularity, so we will finish this section with two results on permutable congruences.

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