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Structural completeness in quasivarieties

Paolo Aglianó, Alex CItkin

Abstract

In this paper we study various forms of (hereditary) structural completeness for quasivarieties of algebras, using mostly algebraic techniques. More specifically we study relative weakly projective algebras and the way they interact with structural completeness in quasivarieties. These ideas are then applied to the study of $C$-structural completeness and $C$-primitivity, through an algebraic generalization of Prucnal's substitution. Finally we study in depth dual i-discriminator quasivarieties in which a particular instance of Prucnal's substitution is used to prove that if each fundamental operation commutes with the i-discriminator, then it is primitive.

Structural completeness in quasivarieties

Abstract

In this paper we study various forms of (hereditary) structural completeness for quasivarieties of algebras, using mostly algebraic techniques. More specifically we study relative weakly projective algebras and the way they interact with structural completeness in quasivarieties. These ideas are then applied to the study of -structural completeness and -primitivity, through an algebraic generalization of Prucnal's substitution. Finally we study in depth dual i-discriminator quasivarieties in which a particular instance of Prucnal's substitution is used to prove that if each fundamental operation commutes with the i-discriminator, then it is primitive.
Paper Structure (20 sections, 72 theorems, 101 equations, 4 figures)

This paper contains 20 sections, 72 theorems, 101 equations, 4 figures.

Table of Contents

  1. Introduction.
  2. Main definitions.
  3. Class operators, quasivarieties an free algebras.
  4. Congruences and subdirectly irreducible algebras.
  5. Relative congruences
  6. Relatively finitely presented algebras.
  7. Relatively weakly projective algebras and primitive quasivarieties.
  8. Structurally complete quasivarieties.
  9. Primitive quasivarieties.
  10. More on structural completeness
  11. $C$-completeness
  12. u-presentability and Prucnal terms
  13. The relative TD-term
  14. Other primitive quasivarieties
  15. Congruence intersection terms
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $\mathsf {K}$ be any class of algebras. Then:

Figures (4)

  • Figure 1:
  • Figure 2:
  • Figure 3: The Fano lattice
  • Figure 4: Simple de Morgan algebras

Theorems & Definitions (129)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • Lemma 2.8
  • ...and 119 more