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The line before Theorem 10.12 says that

"In general $F_K(\overline{X})$ is not isomorphic to a member of K (for example, let K={L} where L is a two-element lattice, then $F_K(\bar{x}, \bar{y}) \notin I(K))$." (The definition of $F_K(\overline{X})$ is shown in page 73.)

Why this is the case? Any example?

Thank you.

=================EDIT====================

This book is freely available as a pdf format : http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf

Due to the lengthy definitions, I provide the link instead of giving full definitions involving the above question. Thanks.

Tim Lee
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  • Would you care to explain what these letters mean? I'm not sure if the set of people who have that book at hand and the set of people willing to answer questions right now intersect non-trivially. – Zavosh Sep 19 '14 at 07:14
  • You ask, "any example?" ...but you already gave an example inside the parentheses. In your example, K contains only the two element lattice L. The free algebra over K on two generators is a 2x2 lattice, which is not isomorphic to L, so it is not isomorphic to any lattice in K. – William DeMeo Sep 19 '14 at 19:18
  • You may find it instructive to try out some small examples in the Universal Algebra Calculator (www.uacalc.org). After you launch the application, try, for example: File -> Built In Algs -> lat2. Then: Tasks -> Free Algebra. You will be prompted for the number if generators. Enter 2, then click "Drawing" and "Go". You will see a 2x2 lattice. When learning this stuff, UACalc can be useful for verifying pencil/paper calculations. – William DeMeo Sep 19 '14 at 19:26
  • Thanks for your reply. I downloaded UACalculator v1.12 and followed your steps. After clicking "Drawing", there are three menus,"Con", "Sub", and "Algebra". By clicking "Sub", I found a four element lattice. Is this the lattice you mentioned? By the way, what does n X n lattice mean? A lattice is defined to be a partially ordered set in which every two elements have a supremum and an infimum. I am new to universal algebra, so I may not know some fundamental things on universal algebra. – Tim Lee Sep 20 '14 at 02:19
  • @TimLee: $n$ means the $n$-element chain, $\times$ means the operation of the direct product of algebras, so $n \times n$ is the lattice which is the direct product of two $n$-element chains. Could you please specify your question? – Random Jack Sep 21 '14 at 18:35
  • @TimLee No, the "Sub" tab draws the lattice of subalgebras of the currently selected algebra. If "Sub" showed a 2x2, that means you're still looking at the 2-element lattice. After following the steps I gave above, you will see two lines in the Algebras list at the bottom of the UACalc window. The first is lat2. The second is the free algebra over lat2 that you constructed using the Tasks menu. Now, from this list of Algebras, select "F(2) over lat2". You can then click the Edit tab and see what the multiplication table is for the free algebra, and draw it using Drawing-->Algebra. – William DeMeo Sep 22 '14 at 01:48
  • @Random Jack, William DeMeo, Thanks for your reply. Let $K={L}$ where L is a two element lattice as shown in the question. In this case, what is $θ_K(X)$ (defined in pp.73)? What are the elements of $F_K(\bar{x},\bar{y})$ and why are they elements of $F_K(\bar{x},\bar{y})$. I used the UACalculator tool, but I am still having difficulty in understanding the nature of $F_K(\overline{X})$. – Tim Lee Sep 23 '14 at 06:23

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