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Questions tagged [universal-algebra]

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).

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).

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Non-roots of unity auxillary constants in a group?

Let $A$ be a set, together with a set $F$ of n-ary operations on A, which may include constants of $A$ as 0-ary operations. A set $G$ of operations on $A$ is said to be auxillary with respect to the algebra $(A,F)$ if $G$ is disjoint from $F$ and…
user107952
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Finite algebraic structure where there is no finite generating set of equations

Let $A$ be an algebra whose carrier set is finite. Must it be the case that there is a finite set of equations which generate all the universally valid equations in that structure? If not, can anyone give me a counterexample?
user107952
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Does $\mathbf{Tos}$ generate $\mathbf{DLat}$ as a variety?

Let $\mathbf{DLat}$ denote the variety of distributive lattices and let $\mathbf{Tos}$ denote the subclass of $\mathbf{DLat}$ consisting of the totally-ordered sets. Question. Does $\mathbf{Tos}$ generate $\mathbf{DLat}$ as a variety?
goblin GONE
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How can I prove that $\ cong(R)‎\cong‎ Id(R)$?

If $R$ is an arbitrary ring, $\ cong(R)$ is the set of all congruence of $R$ and $Id(R)$ is the set of all Ideals of $R$, How can I prove that $\ cong(R)‎\cong‎ Id(R)$?
Jamal
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Are there any preservation theorems for quotients of subalgebras?

Let $X$ denote an algebraic structure. Then: Every subalgebra of $X$ satisfies each quasi-identity that is satisfied by $X$. In other words, taking subalgebras preserves quasi-identities. Every quotient of $X$ satisfies every…
goblin GONE
  • 67,744
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Clone of operations acting bicentrally

Let $(A,F)$ be an algebra, let $F^{\star}$ be the centralizer of $F$ and $F^{\star\star}:=(F^{\star})^{\star}$ the bi-centralizer. Let $[F]$ denote the clone of operations generated by $F$. We say that: 1) $F$ acts bicentrally on $A$ when…
Danae Kissinger
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The concept of K-free algebras

This is Definition 10.9 of the book "A Course in Universal Algebra" by Burris and Sankappanavar (page 73, Millennium Edition). http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf Definition 10.9 Let $K$ be a family of algebras of…
Tim Lee
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Basic question about the definition of a variety (in universal algebra)

According to wikipedia, A variety is a class of algebraic structures of the same signature that is closed under the taking of homomorphic images, subalgebras and (direct) products. Isn't the stipulation about subalgebras redundant? I think…
goblin GONE
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Are there any new identities when we go from subtraction to subtraction with a nonzero constant?

This is the subtraction counterpart to my previous universal algebra question on addition with a nonzero constant, here: No simplifying identities for any single nonzero number under addition.. I know that the structure $(\mathbb{R};-)$ of the…
user107952
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A characterization of the identities which are equivalent to the trivial identity

This is a follow-up to my previous question, here: Is only the commutative identity equivalent to the commutative identity?. As in that question, let our signature be that of a single binary operation symbol $+$, which is just a symbol for an…
user107952
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Atoms and coatoms of the partial order of equations

Let our signature be that of a single binary operation symbol $*$. Consider the set of equations in that signature. I define a preorder on that set by saying $E \geq E'$ if and only if the equation $E$ implies the equation $E'$. By quotienting out…
user107952
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Is this identity equivalent to the constant identity?

Let our signature be that of a single binary operation $*$. I define $*$ to satisfy the constant identity if $x*y=z*w$. But, I am interested in another identity, this one: $x*y=y*z$. I want to know if it is equivalent to the constant identity. I…
user107952
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Is the equational theory generated by this set of equations the same as the set itself?

This is related somewhat to my previous question, here: What is a finite equational basis for this equational theory?. As before, our signature is a single binary operation $*$. Also as before, let $E$ be the set of all equations $s=t$, where $s$…
user107952
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What is a finite equational basis for this equational theory?

Let our signature be a single binary operation $*$. Let $E$ be the set of all equations $s=t$ such that the set (the set, not the multiset) of variables in the term $s$ is the same as the set of variables in the term $t$. So, for example, the…
user107952
  • 20,508
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The commutator is meet semidistributive in the second variable

Let A be an algebra and let $\alpha,\beta,\gamma$ congruence in A. How can I prove that the commutator is meet semidistributive in the second variable ( i.e. $[\beta,\alpha]=[\beta,\gamma]$ implies $[\beta,\alpha]=[\beta,\gamma\vee\alpha]$ ) ? Def.…
Seurat
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