# Tagged Questions

The study of algebraic structures and properties applying to large classes of such structures. For example, ideas from group theory and ring theory are extended and considered for structures with other signatures (systems of basic or fundamental operations).

232 views

### Is the equational theory of commutative vN regular rings decidable?

I wanted to check whether $A(x,y):=\frac{xy}{x+y}$ is an associative operation in every commutative vN regular ring. Now $A(-1,A(1,1))=A(-1,\frac{1}{2})=1\neq 0 =A(0,1)=A(A(-1,1),1)$. On the other ...
278 views

### Example of a non-finitely based variety with explicit set of defining identities

There are many examples of non-finitely based varieties. In a finite signature, is there an example of such variety with a known explicit set of identities?
741 views

### Examples of algebras satisfying (a+b)(c+d)=ac+bd

Is there a known example of an algebra $(A, +, \cdot)$ with two binary commutative (see P.S below) and idempotent operations $+$ and $\cdot$ satisfying the identity $(a+b)(c+d)=ac+bd$? Actually I ...
632 views

### Natural associative law for a ternary “group”?

Suppose one were to define a group-like structure based on a set $G$ with a ternary (rather than binary) operator $g( a, b, c ) = \left< a, b, c \right>$. One possible definition for the ...
268 views

### The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$. One important example of left distributive algebras arises ...
175 views

109 views

### Which algebraic theories have the property that $\mid$ is antisymmetric for all free algebras?

Let $T$ denote an algebraic theory. Terminological Question. Let $X$ denote a $T$-algebra. Is there a name for the preorder $\mid$ defined on $X$ by asserting that $a \mid b$ iff there is a term ...
178 views

116 views

### A generalization of quasi-identities

In universal algebra, a variety is axiomatized by identities $t \approx s$ between terms $t$ and $s$. More general are quasi-varieties that are axiomatized by quasi-identities of the form u_1 \...
814 views

### Commutative associative rational binary operations

What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.) Feel free to re-tag if you can think of ...
158 views

### How to prove that a binary relation is a strongly rigid relation? i.e. Polρ only contains projections

I first quote a definition from Clone theory in Universal Algebra: A binary relation $\rho$ on a set U is strongly rigid if every universal algebra on U such that $\rho$ is a subuniverse of its square ...
Last month I proved that some category $\mathbf C$ that I happen to care about is isomorphic to the Eilenberg-Moore category for a monad on the category of bounded posets $\mathbf{BPos}$. I know from ...