Basic Axiomatic Definitions in Categories and Allegories (Freyd and Scedrov) I'd like to ask about the Basic Definitions given at the very beginning of the Categories and Allegories. Some aspects of the text are idiosyncratic, so first I'll quote from the text:
1.1 BASIC DEFINITIONS
The theory of CATEGORIES is given by two unary operations and a binary partial operation. In most contexts lower-case variables are used for 'individuals' which are called morphisms or maps. The values of the operations are denoted and pronounced as: 
\begin{align}
\square x &&\mbox{the source of x,}\\
x \square &&\mbox{the target x,} \\
xy && \mbox{the composition of x and y.}
\end{align}
The axioms:
\begin{align}
xy && \mbox{is defined iff} && x\square = \square y, \\
(\square x)\square = \square x && \mbox{and} && \square(x \square)=x\square, \\
(\square x)x = x && \mbox{and} &&  x(x\square)=x, \\
\square(xy) = \square(x(\square y)) && \mbox{and} && (xy)\square = ((x\square)y)\square, \\
x(yz) = (xy)z.
\end{align}
1.11 The ordinary equality sign = will be used only in the symmetric sense, to wit: if either side is defined then so is the other and they are equal. A theory, such as this, built on an ordered list of partial operations, the domain of definition of each given by equations in the previous, and with all other axioms equational, is called an ESSENTIALLY ALGEBRAIC THEORY.
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My first question is: What is a binary partial operation? I have tried looking elsewhere for the definition, but most of what I see (online) is given within the context of technical discussions, mired in further terminology. Is there an accessible route to this notion?
Second: Regarding the axioms, I am having difficulty interpreting some of these. For example, how to view $(\square x)x = x$ ? Well, $\square x$ is defined as the source (domain) of the morphism $x$. But what does it mean for this expression to sit in front of $x$. Perhaps the solution is to suspend preconceptions (like set theory and functions) and continue reading. 
Finally: As I read the definition of essentially algebraic theory, I'm trying to see what type of theory would not qualify as essentially algebraic. Similar to the first question, is there an accessible account of what the enterprise here is?
I've read elsewhere that the text was engineered to be self-contained. Stated prerequisites (beyond sophistication, surely) are "some basic topology and algebra…" That's my modest background (munkres, herstein/artin). Perhaps I should interpret that as a hint to just read on, notwithstanding undefined terms like "partial operation", which are then used in tightly packed definitions (like "essentially algebraic theory"). 
Thanks for any help. 
 A: A binary operation on a class $S$ is a function $S \times S \to S$. A binary partial operation is a partial function $S \times S \to S$. That is, it can be expressed as a function $D \to S$ for some subclass $D \subseteq S \times S$.
For example, multiplication is a binary operator and division is a binary partial operator on the real numbers.

This is the "arrows only" definition of a category. In this formulation, there is no such thing as an "object" at the lowest level; it's all just arrows. As such, $\square x$ is an arrow, and therefore $(\square x)x$ is the composition of two arrows $\square x$ and $x$.
In this formalism, when we introduce the notion of "object", we define an object to be an identity morphism. The axioms show that $\square x$ is, in fact, an identity morphism.

Essentially algebraic theories are a generalization of universal algebras; it's flavor is still that they are still theories of functions and are axiomatized in terms of algebraic identities.
The theory of a field is not essentially algebraic: the key flaw in the usual formulation is the need to define the class of all nonzero numbers (e.g. so as to assert that said class is the domain of the inversion operator), which can't be done via equations.
