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An embedding theorem for partial algebras and the free completion of a free partial algebra within a primitive class

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Abstract

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let U f denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some U f -algebra. For any U-algebraAsetU A =U i εI({i}×(A K i—domf i A)), where (K i) i εI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, U f ) be similarly defined). Then for every nonempty setM there exists a setN satisfyingMN such that there exists a bijective mapping σ:U F(N, U)NM satisfying σ((i, α)) ∉ α(K i ) for all (i, α) ∈U F (N, U), and, for the structureg=(g i)iεI defined by ,g i : =f i F(N, U) ∪ {(α, σ((i, α))) | (i, α ∈U F(N, U)} id M induces an isomorphism betweenF(M, U f ), and (F(N, U)g).

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References

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Author information

Correspondence to P. Burmeister.

Additional information

To Reinhold Baer on his seventieth birthday

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Burmeister, P. An embedding theorem for partial algebras and the free completion of a free partial algebra within a primitive class. Algebra Univ. 3, 271–279 (1973). https://doi.org/10.1007/BF02945128

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Keywords

  • Weak Sense
  • Ordinal Number
  • Homomorphic Image
  • Relative Algebra
  • Surjective Homomorphism