$\newcommand{\o}[0]{\mathsf{orth}}$No, these equations do not yield the complete theory of the orthocenter.
The identity
$$\o(\o(t,u,v),\o(t,u,w),u) = \o(\o(t,u,v),\o(t,v,w),v)$$
holds for the orthocenter (X(4)) but not for X(74) (the isogonal conjugate of the Euler infinity point), even though both satisfy involutory identities like
$$v=\o(t,u,\o(t,u,v))$$
I found the first identity and tested both using Mathematica, which can set up the functions quickly as follows:
avg[a_, b_, c_, u_, v_, w_] := (a u + b v + c w)/(a + b + c);
bary[f_, u_, v_, w_] := avg[f[u,v,w], f[v,w,u], f[w,u,v], u, v, w];
cosA[u_, v_, w_] := (v-u).(w-u) / Sqrt[((v-u).(v-u)) ((w-u).(w-u))];
f[u_, v_, w_] := 1/((u-v).(u-w));
g[u_, v_, w_] := Sqrt[(v-w).(v-w)] / (cosA[u,v,w] - 2 cosA[v,w,u] cosA[w,u,v]);
center4[u_, v_, w_] := bary[f, u, v, w];
center74[u_, v_, w_] := bary[g, u, v, w];
Then the following code tests the identity for the orthocenter algebraically, and for the X(74) center numerically:
Algebra = {t -> {tx, ty}, u -> {ux, uy}, v -> {vx, vy}, w -> {wx, wy}};
Example = {t -> {0, 1}, u -> {2, 4}, v -> {-3, 5}, w -> {-2, 1}};
FourVariable[c_] := c[c[t,u,v], c[t,u,w], u] == c[c[t,u,v], c[t,v,w], v];
{FourVariable[center4] /. Algebra // Simplify, FourVariable[center74] /. Example}
and the tests return True and False respectively.
There is a simple characterization that $\def\LL{\mathcal L}\LL$ is nowhere negative iff all $\LL$-theories have the disjunction property:
Proposition. For any $\LL\subseteq\mathrm{FO}$, the following are equivalent:
- $\LL$ is nowhere negative.
- For every $\LL$-theory $T$ and every finite set of $\LL$-formulas $\{\phi_i:i\in I\}$,
$$T\models\bigvee_{i\in I}\phi_i\implies\exists i\in I\:T\models\phi_i.$$
Note that this is equivalent to the special cases where $|I|$ is either $0$ (which amount to the consistency of $T$) or $2$.
- The same for finite theories $T$.
Proof.
$1\to2$: Let $A$ be a model whose theory is $T$. Then $A\models\bigvee_{i\in I}\phi_i$ implies $A\models\phi_i$ for some $i\in I$ by the definition of satisfaction of disjunctions.
$2\to3$ is trivial.
$3\to1$: For any $\LL$-theory $T$, the $\mathrm{FO}$-theory
$$T^*=T+\{\neg\phi:\phi\in\LL,T\not\models\phi\}$$
is consistent: if it were not, there would be a finite $T'\subseteq T$ and a finite set $\{\phi_i:i\in I\}$ of $\LL$-sentences such that $T\not\models\phi_i$ (hence $T'\not\models\phi_i$) for each $i\in I$, and $T'\cup\{\neg\phi_i:i\in I\}$ is inconsistent. But then $T'\models\bigvee_{i\in I}\phi_i$, contradicting the disjunction property of $T'$.
Then any model of $T^*$ has the property that its $\LL$-theory consists exactly of the $\LL$-consequences of $T$. QED
Note that the only properties of FO we used here is that it is closed under Boolean connectives, and obeys the compactness theorem.
It follows immediately from the Proposition that the property of being nowhere negative is stable under unions of chains after all, hence by Zorn’s lemma, maximal nowhere negative sublogics of FO exist (as long as we fix the language so that the class of all FO sentences is a set).
While it is neither here nor there, let me mention another fact from the comments: the equational logic example generalizes to the logic $\LL\subseteq\mathrm{FO}$ of sentences built from atomic formulas by means of $\exists$, $\forall$, and $\land$. The fact that all $\LL$-theories have the disjunction property, hence $\LL$ is nowhere negative, is a consequence of the property that for every family of models $\{A_i:i\in I\}$ (where $I$ may be empty) and every $\LL$-sentence $\phi$,
$$\prod_{i\in I}A_i\models\phi\iff\forall i\in I\:A_i\models\phi.$$
The left-to-right implication follows from the fact that all $\LL$-sentences are positive (monotone), hence preserved under surjective homomorphisms (incuding projections from products); the right-to-left implication follows from the fact that $\LL$-sentences are strict Horn sentences. I do not know whether $\LL$ is maximal.
Best Answer
Let me copy Definition 4.1 of
Libor Barto, Jakub Oprsal, Michael Pinsker
The wonderland of reflections
Israel Journal of Mathematics 223 (2018), 363-398
Defn. 4.1 Let $\mathbf{A}$ be an algebra with signature $\tau$. Let $B$ be a set, and let $h_1\colon B\to A$ and $h_2\colon A\to B$ be functions. We define a $\tau$-algebra $\mathbf{B}$ with domain $B$ as follows: for every operation $f(x_1,\ldots,x_n)$ of $A$, $\mathbf{B}$ has the operation $$(x_1,\ldots,x_n) \mapsto h_2(f(h_1(x_1),\ldots, h_1(x_n))).$$ We call $\mathbf{B}$ a reflection of $\mathbf{A}$. If $h_2 \circ h_1$ is the identity function on $B$, then we say that $\mathbf{B}$ is a retraction of $\mathbf{A}$.
The construction in the problem statement above is the special case of a reflection where $B=\textrm{im}(f)$, $h_2=f$, and $h_1$ = the inclusion of $B$ into $A$. In this case, the algebra $\mathbf{B}$ of Definition 4.1 is what would be called $\mathbf{A}_{h_2}$ in the problem statement.
The idempotent construction in the problem statement is exactly what is meant in Theorem 4.1 by a retraction. (If $f$ is idempotent in the problem statement, let $B=\textrm{im}(f)$, $h_2 = f$, $h_1$ = the inclusion of $B$ into $A$ to see that $\mathbf{A}_f$ is what is called a retraction of $\mathbf{A}$ in the wonderland paper. Conversely, if $\mathbf{B}$ is a retraction of $\mathbf{A}$ in the sense of Definition 4.1, let $f = h_1\circ h_2$ in the problem statement and notice that $h_1\colon \mathbf{B}\to \mathbf{A}_f$ is an isomorphism.)
Now let me summarize some definitions from the wonderland paper. A term has height 0 if it is a variable and has height 1 if it is a fundamental operation applied to variables. A term has height at most 1 if its height is 0 or 1. An identity $s\approx t$ has height 1 if both $s$ and $t$ have height 1, and has height at most 1 if both $s$ and $t$ have height at most 1.
The wonderland result that is relevant to this problem is:
Corollary 5.4: Let $K$ be a nonempty class of algebras of the same signature $\tau$.
(i) $K$ is closed under reflections and products if and only if $K$ is the class of models of some set of $\tau$-identities of height 1.
(ii) $K$ is closed under retractions and products if and only if $K$ is the class of models of some set of $\tau$-identities of height at most 1.
With this background, let me address the questions in the problem.
Is there a $\mathbb{K}$ such that $F_i(\mathbb{K})\subsetneq F(\mathbb{K})$?
Yes, let $\mathbb{K}$ be the class of semilattices. It can be shown that the associative law is not a consequence of height 1 identities, so $\mathbb{K}\subsetneq F_i(\mathbb{K})$. Also, the idempotent law $x\wedge x\approx x$ has height at most 1, so it will be satisfied in $F_i(\mathbb{K})$. (This fact is easy to prove from the definitions, you don't need to cite the wonderland paper.) The class $F(\mathbb{K})$ does not satisfy $x\wedge x\approx x$, because this class contains nontrivial algebras of size $>1$ that have constant multiplication. To see this, choose a semilattice $\mathbf{A}$ and a nonconstant function $f\colon A\to A$ such that $\textrm{im}(f)$ is a proper subsemilattice of $\mathbf{A}$ and $f\circ f$ is constant. In this situation $\mathbf{A}_f$ will have size $>1$ and have constant multiplication, so it will not satisfy the idempotent law, so it will lie in $F(\mathbb{K})-F_i(\mathbb{K})$.
Is the equational theory of $F_i(\mathbb{K})$ the (deductive closure of the) set of basic equations true in each element of $\mathbb{K}$?
Yes, as indicated by Corollary 5.4 of the wonderland paper.
Related remarks.
Remark 1. The phrase 'basic equations' goes back to
Kelly, David
Basic equations: word problems and Mal’cev conditions.
Abstract 701-08-04, AMS Notices 20 (1972) A-54.
For Kelly, a term is basic if it is a variable, a constant, or a fundamental operation symbol applied to variables and constants. An identity is a basic equation if both sides are basic. This is close to but different from what is meant in the problem on this page.
Remark 2. The part of Corollary 5.4 that concerns retractions was observed earlier in
Taylor, Walter
Simple equations on real intervals.
Algebra Universalis 61 (2009), no. 2, 213-226.
I summarize Taylor's Theorem 2.1 as follows:
Theorem 2.1. Let $V$ be a variety Then $V$ is closed under set-retractions iff $V$ is definable by simple equations.
Taylor's terminology is a little different than that of the wonderland paper, but the result is the same in the case of retractions. (Simple = height at most 1.)
You can find a extended discussion of Theorem 2.1 in Volume 2, Section 6.7, of the encyclopedia
Ralph S. Freese, Ralph N. McKenzie, George F. McNulty, Walter F. Taylor
Algebras, Lattices, Varieties
Mathematical Surveys and Monographs, American Mathematical Society, v. 268, 2022.
[In particular, FMMT show in their Theorem 6.61 that if $\mathbb{K}$ is the class of semilattices, then $\mathbb{K}\subsetneq F_i(\mathbb{K})$, because the latter does not satisfy the associative law.]
Remark 3. The wonderland paper was written primarily to develop tools to study the Constraint Satisfaction Problem for $\aleph_0$-categorical structures.