 # Montely mathematics: Some Algebra(s)

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Initially, I would like to congratulate Miller for quickly answering both questions last month. Now, to mathematics. The topic for this month is a hodgepodge of linear algebra, field theory, and universal algebra. I think that the wall of text last time scared some people off. So, I tried to be as concise as possible this time.

Note: Due to life being crazy, this post will go up without as much editing as it deserves. I apologize for this, and will appreciate any corrections.

Notation:

I believe that all of my notation here is fairly standard with the exception of scalar multiplication. I denote it as follows: $( S^{\mathbb{V}}_{c} )_{c \in F}$. The vector passed to $S^{\mathbb{V}}_c$ is scaled by c, where $c \in F$.

Definitions:

1. Algebra:

An algebra is an ordered pair $\mathbb{A} = ( A, \mathcal{F} )$ where A is the universe, i.e. what elements exist in $\mathbb{A}$, $\mathcal{F}$ is a family of functions. These functions are call fundamental operations. The fundamental operations are closed, i.e. their range is a subset of A.

1. Congruence lattice:

A congruence lattice is a lattice who’s universe is the set of all congruences on $\mathbb{A}$ and meets and joins are calculated in the usual way over equivalence relations.

1. Field:

A field, $\mathbb{F} = (F; +^{\mathbb{F}}, *^{\mathbb{F}}, -^{\mathbb{F}}, ^{-1^{\mathbb{F}}}, 1^{\mathbb{F}}, 0^{\mathbb{F}})$, is an algebraic structure with an additive and multiplicative operation such that the field axioms are satisfied.

1. Vector Space:

Due to time constraints, I am going to link to a definition: http://en.wikipedia.org/wiki/Vector_space#Definition

1. Operation Table:

An example of a operation table is the addition or multiplication table. An operation table is a table that shows you what the output of a function is based on the input.

Theorems:

1. Meet and join for relations
2. Let $\theta_{1}$ and $\theta_{2}$ be an equivalence relations for some algebra, then the usual meet and join are calculated as follows:

• $\theta_{1} \wedge \theta_{2} = \theta_{1} \cap \theta_{2}$
• $\theta_{1} \vee \theta_{2} = \theta_{1} \cup ( \theta_{1} \circ \theta_{2} ) \cup ( \theta_{1} \circ \theta_{2} \circ \theta_{1} ) \cup ( \theta_{1} \circ \theta_{2} \circ \theta_{1} \circ \theta_{2} ) \cup ...$

Questions:

1. Suppose $F = \{ 0, 1, \alpha, \alpha + 1 \}$ and $\mathbb{F} = (F; +^{\mathbb{F}}, *^{\mathbb{F}}, ^{-1^{\mathbb{F}}},-^{\mathbb{F}}, 1^{\mathbb{F}}, 0^{\mathbb{F}})$, i.e $\mathbb{F}$ is a field. Define $\alpha^{2} = \alpha + 1$ and $\alpha^{2} + \alpha + 1 = 0$. What do the operation tables for $+^{\mathbb{F}}$ and $*^{\mathbb{F}}$ look like.
2. Let $F = \{ 0, 1, \alpha, \alpha + 1 \}$, and $\mathbb{V} = (V; +^{\mathbb{V}}, (S^{\mathbb{V}}_{c})_{c\in F}, -^{\mathbb{V}}, 0^{\mathbb{V}})$ be a vector space where $V=F\times F$. What is the structure of $\mathbb{C} \bold{on} ( \mathbb{V} )$?

Let me assure you non-mathematician reader, you most certainly can answer the first question. I’m not saying it will be necessarily obvious, but you have the skills and reasoning abilities to solve it! The second question, granted, may require some familiarity with relations, lattices, and vector spaces. In general, let me encourage you to ask questions! I, or someone else, will be happy to help you solve these problems.