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Płonka Adjunction

Juan Climent Vidal, Enric Cosme Llópez

Abstract

For a signature $Σ$ and its subsignature $Σ^{\neq 0}$ without $0$-ary operation symbols, we prove (1) that there are strong Lawvere adjoint cylinders between the category $\mathsf{Ssl}$, of sup-semilattices, and the categories $\int^{\mathsf{Ssl}}\mathrm{Isys}_Σ$, of sup-semilattice inductive systems of $Σ$-algebras, and $\int^{\mathsf{Ssl}}\mathrm{Isys}_{Σ^{\neq 0}}$, of sup-semilattice inductive systems of $Σ^{\neq 0}$-algebras; (2) that there exists an adjunction between $\mathsf{Ssl}$ and the category $\mathsf{Alg}(Σ^{\neq 0})$, of $Σ^{\neq 0}$-algebras; (3) that there exists an adjunction between the categories $\mathsf{Ssl}$ and $\mathsf{Lnb}$, the category of left normal bands; (4) after defining and stating several technical results on the category $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$, of Płonka $Σ^{\neq 0}$-algebras, and defining functors $J_{Σ^{\neq 0}}$ from $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$ to $\mathsf{Alg}(Σ^{\neq 0})\otimes\mathsf{Lnb}$, the tensor product of $\mathsf{Alg}(Σ^{\neq 0})$ and $\mathsf{Lnb}$, and $P_{Σ^{\neq 0}}$ from $\mathsf{Alg}(Σ^{\neq 0})\otimes\mathsf{Lnb}$ to $\mathsf{Alg}(Σ^{\neq 0})$, we prove that $P_{Σ^{\neq 0}}\circ J_{Σ^{\neq 0}}$ has a left adjoint; finally, (5) after defining a functor $\mathrm{Is}_{Σ^{\neq 0}}$ from $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$ to $\int^{\mathsf{Ssl}}\mathrm{Isys}_{Σ^{\neq 0}}$ we prove the main result of this paper: that $\mathrm{Is}_{Σ^{\neq 0}}$ has a left adjoint $\mbox{\upshape{Pł}}_{Σ^{\neq 0}}$, which is the Płonka sum.

Płonka Adjunction

Abstract

For a signature and its subsignature without -ary operation symbols, we prove (1) that there are strong Lawvere adjoint cylinders between the category , of sup-semilattices, and the categories , of sup-semilattice inductive systems of -algebras, and , of sup-semilattice inductive systems of -algebras; (2) that there exists an adjunction between and the category , of -algebras; (3) that there exists an adjunction between the categories and , the category of left normal bands; (4) after defining and stating several technical results on the category , of Płonka -algebras, and defining functors from to , the tensor product of and , and from to , we prove that has a left adjoint; finally, (5) after defining a functor from to we prove the main result of this paper: that has a left adjoint , which is the Płonka sum.
Paper Structure (7 sections, 38 theorems, 140 equations, 2 figures)

This paper contains 7 sections, 38 theorems, 140 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On the strong Lawvere adjoint cylinder between $\int^{\mathsf{Ssl}}\mathrm{Isys_{\Sigma}}$ and $\mathsf{Ssl}$
  4. On the relationship between $\mathsf{Ssl}$ and $\mathsf{Alg}(\Sigma^{\neq 0})$
  5. On the category $\mathsf{Lnb}$ of left normal and its relationship with the category $\mathsf{Ssl}$
  6. On the category of Płonka $\Sigma^{\neq 0}$-algebras
  7. On the relationship between $\hbox{\sffamily{\upshape{P{\l}Alg}}}(\Sigma^{\neq 0})$ and $\int^{\mathsf{Ssl}}\mathrm{Isys}_{\Sigma^{\neq 0}}$: Płonka adjunction

Key Result

Proposition 2.7

Let $X$ be a set. Then, for every $P\in W_{\Sigma}(X)$, we have that $P$ is a term with variables in $X$ if, and only if, $P = (x)$, for a unique $x\in X$, or $P = (\sigma)$, for a unique $\sigma\in\Sigma_{n}$, or $P = (\sigma)\curlywedge\text{\Large $\curlywedge$}_{j\in n}P_{j}$, for a unique $n\in

Figures (2)

  • Figure 1: The Universal Property in Proposition \ref{['LAdj']}.
  • Figure 2: A general overview of the paper.

Theorems & Definitions (135)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 125 more