Mathematics Stack Exchange
Mathematics Stack Exchange

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).

891 questions
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A Birkhoff theorem on K-free algebras

Let $F_K(\overline{X})$ be the $K$-free algebra over $\overline{X}$. I want to prove that $F_K(\overline{X})\in ISP(K)$. I have already proved that $F_K(\overline{X})\in IP_SIS(K)$. Since $P_S\leq SP$, we have $F_K(\overline{X})\in ISPIS(K)$. How…
Danae Kissinger
  • 2,143
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What must we require of an algebraic theory for the inclusion of generators map to be injective?

Let $T$ denote an algebraic theory. Then given a free $T$-algebra $F(k)$, the inclusion of generators map $\eta : k \rightarrow U(F(k))$ is usually injective, in practice. It doesn't have to be, though; consider the case where $T$ proves the…
goblin GONE
  • 67,744
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If a finite $T$-algebra only satisfies the identities of $T$ and no others, is it a free object?

My original question was the following: Let $T$ denote an algebraic theory and suppose $X$ is a $T$-algebra. If for every identity $\eta$ in the language of $T$ we have that $(X \models \eta) \rightarrow (T \vdash \eta),$ is it necessarily the case…
goblin GONE
  • 67,744
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3 answers

n-ary derived operation in universal algebra

I've just come across the definition of the n-ary derived operation, namely that starting with an operational type $(\Omega, \alpha)$, set $ X_n = (x_1, ... , x_n) $ and $ \Omega$-structure $A$, we can define a function $ t_A : A^n \to A $ for each…
TRY
  • 397
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1 answer

Smallest cardinality of finite algebraic structure whose quasi-equational theory is not finitely based?

I understand that, for the set $\{0,1\}$ and any finite number of constants and operations on that set, the resulting algebraic structure has a finite basis of identities. But, does there exist a finite algebraic structure on $\{0,1\}$ whose…
user107952
  • 20,508
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A Mal'cev algebra $\mathbf{A}$ have typ$\{\mathbf{A}\} \subseteq \{\mathbf{2},\mathbf{3}\}$

A Mal'cev algebra is an algebra $\mathbf{A}$ with a ternary term $t$ such that $\mathbf{A} \models t(x,x,y) \equiv y, t(x,y,y) \equiv x$. For the remaining of the question, refer to the definitions and results contained in Hobby and McKenzie's book…
Mockingbird
  • 983
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Free algebra over the intersection of varieties

I am wondering if given to varieties $V,W$ in the sense of Universal algebra once could describe free algebras over $V\cap W$ in some meaningful way? Now what I mean by meaningful is not even completely clear to me. I would like to use this…
TdotA
  • 400
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Every variety has a simple algebra -- proof

I'm having a hard time understanding the proof (as presented for example in A Course in Universal Algebra of the GTM series) of the theorem by Magari which states that a variety $V$ with a nontrivial member always contains a simple algebra. The…
Mockingbird
  • 983
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What is a basis for the common equational identities of $(\mathbb{R};*,0)$, $(\mathbb{R};*,1)$, and $(\mathbb{R};*,-1)$?

Consider the three algebraic structures $(\mathbb{R};*,0)$, $(\mathbb{R};*,1)$, and $(\mathbb{R};*,-1)$, where $*$ denotes multiplication. What is a basis for the equational identities those structures have in common? I conjecture that these…
user107952
  • 20,508
2
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1 answer

Is only the commutative identity equivalent to the commutative identity?

Let our signature be that of a single binary operation $+$. Suppose I have an equational identity $E$ such that $E$ is equivalent to the commutative identity $x+y=y+x$. In other words, $E$ implies and is implied by the commutative identity. Must $E$…
user107952
  • 20,508
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1 answer

Basis for the equational identities of the algebraic structure $(\mathbb{R};-,abs)$

Consider the algebraic structure $(\mathbb{R};-,abs)$, where $-$ is the additive inverse unary function, and $abs$ is the absolute value function. What is a basis for the equational identities of that structure? I conjecture that just these three…
user107952
  • 20,508
2
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2 answers

Is there a clone on a three element set which is not finitely generated?

Let 3 be the three element set $\{0,1,2\}$. Is there a clone (in the sense of universal algebra) on 3 which is not finitely generated? I know that every clone on a two element set is finitely generated, so what about a three element set? And if the…
user107952
  • 20,508
2
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1 answer

Under which conditions are finitely presented algebras finite?

Let V be a nontrivial variety with a finite signature. Under which conditions (if any) is every algebra finite which is finitely presentable in V?
Ohbi
  • 85
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1 answer

Is subtraction term-definable from addition in the reals?

In the structure $(\mathbb{R};-)$, where $-$ is binary subtraction, addition is a term function in it, like this: $x+y=x-((y-y)-y)$. I want to know if subtraction is a term function in the structure $(\mathbb{R};+)$. I know that it is definable from…
user107952
  • 20,508
2
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1 answer

Is the set of identities of a single binary operation densely ordered under this strict partial order?

This is somewhat similar to my previous question, here: Is the partial order of equational theories of a single binary operation dense?, but slightly different. Consider the set $S$ of identities of a single binary operation $\{*\}$. So, for…
user107952
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