List of algebraic structures

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In universal algebra, a branch of pure mathematics, an algebraic structure consists of a set closed under one or more operations, satisfying a number of axioms, including none. Abstract algebra is primarily the study of algebraic structures and their properties.

This definition of an algebraic structure should not be taken as restrictive. Anything that satisfies the axioms defining a structure is an instance of that structure, regardless of how many other axioms that instance happens to satisfy. For example, all groups are also semigroups and magmas.

Structures are listed below in approximate order of increasing complexity as follows:

• Structures that are varieties precede those that are not;
• Simple structures built on one set, the universe S, precede composite ones built on two;
• If A and B are two sets making up a composite structure, that structure may include functions of the form AxA→B or AxB→A.
• Structures are then ordered by the number and arities of the operations they contain. No structure mentioned in this entry requires an operation whose arity exceeds 2.

If structure B is under structure A and more indented, then all theorems of A are theorems of B; the converse usually does not hold.

Contents 1 Structures that are varieties 1.1 Simple structures 1.2 Group-like structures 1.3 Lattices 1.4 Ring-like structures 1.5 Modules and Algebras 2 Structures that are not varieties 2.1 Arithmetics 2.2 Lattices that are not varieties 2.3 Field-like structures 2.4 Vector spaces that are not varieties 2.5 Multilinear algebras 3 Examples 3.1 Arithmetics 3.2 Group-like structures 3.3 Lattices 3.4 Ring-like structures 3.5 Field-like structures 4 Allowing additional structure 5 See also 6 References 7 External links

[edit] Structures that are varieties

Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over a set that is part of the definition of the structure. Hence identities contain no sentential connectives, existentially quantified variables, or relations of any kind other than equality and the allowed operations.

If the axioms defining an algebraic structure are all identities, or can be recast as identities, the structure is a variety (not to be confused with algebraic variety in the sense of algebraic geometry). For example, the Boolean nonidentity α≤β can always be recast as the Boolean identity αβ=α.

[edit] Simple structures

No binary operation.

• Set: a degenerate algebraic structure having no operations.
• Pointed set: S has one or more distinguished elements, often 0, 1, or both. While pointed sets are near-trivial, they lead to pointed spaces, which are not.
• Unary system: S and a single unary operation over S.
• Pointed unary system: a unary system with S a pointed set.

[edit] Group-like structures

One binary operation, denoted by concatenation. For monoids, loops, and sloops, S is a pointed set.

• Magma or groupoid: S is closed under a single binary operation.
  • Steiner magma: A commutative magma satisfying x(xy) = y.
    • Squag: an idempotent Steiner magma.
    • Sloop: a Steiner magma with distinguished element 1, such that xx = 1.
  • Order algebra: a magma idempotent in the following wide sense. Either but not both of two (non)contiguous instances of the subformula x may be deleted. Conversely, x may be duplicated freely in the same formula.
  • Semigroup: an associative magma.
    • Monoid: a unital semigroup.
      • Group: a monoid with a unary operation, inverse, giving rise to an inverse element equal to the identity element.
        • Abelian group: a commutative group.
        • Group with operators: a group with a set of unary operations over S, with each unary operation distributing over the group operation.
      • Logic algebra: a commutative monoid (equivalently, a unital semilattice) with a unary operation, complementation, denoted by enclosing its argument in parentheses (the group operation not requiring parentheses because it associates). Complementation gives rise to an inverse element that is the complement of the identity element, so that (x)x=(1) is an axiom. The identity and inverse elements bound S in the lattice sense. Logic algebras also satisfy ((x))=x and x(1)=(1).
        • MV-algebra: a logic algebra satisfying the axiom ((x)y)y = ((y)x)x.
        • Boundary algebra: a logic algebra satisfying the axiom (xy)y = (x)y, from which follow ((x))=x, x(1)=(1), ((x)y)y = xy, and the idempotence of the group operation. Also an order algebra.
    • Band: an idempotent semigroup.
      • Semilattice: a commutative band. The group operation is one of meet or join.
      • Normal band: a band satisfying the axiom xyzx = xzyx.
      • Rectangular band: a band satisfying the axiom xyz = xz.

Two binary operations.

• Hoop: a commutative monoid with an additional binary operation, denoted by infix →, satisfying the axioms x→(y→z) = (xy)→z, x→x = 1, and (x→y)x = (y→x)y.

Three binary operations. Quasigroups are listed here, despite their having 3 binary operations, because they are (nonassociative) magmas. Quasigroups feature 3 binary operations only because establishing the quasigroup cancellation property by means of identities alone requires two binary operations in addition to the group operation.

• Quasigroup: a cancellative magma. Equivalently, ∀x,y∈S, ∃a,b∈S, such that xa = y and bx = y.
  • Loop: a unital quasigroup with a unary operation, inverse.
    • Moufang loop: a loop satisfying a weakened form of associativity, (zx)(yz) = z(xy)z.
      • Group: an associative loop.

[edit] Lattices

Two binary operations, meet (infix ∧)and join (infix ∨), that can be proved idempotent, and are connected by the absorption law. Interchanging all meets and joins preserves truth; this is the duality principle.

N.B. "Lattice" has another meaning unrelated to this section, namely a discrete subgroup of the real vector space Rⁿ that spans Rⁿ. See lattice (group).

• Latticoid: meet and join commute but do not associate.
• Skew lattice: meet and join associate but do not commute.
• Lattice: a commutative skew lattice and an associative latticoid. A lattice is both a meet and join semilattice.
  • Bounded lattice: a lattice with two distinguished elements, the greatest (1) and the least element (0), such that x∨1=1 and x∨0=x. Dualizing requires interchanging 0 and 1. A bounded lattice is a pointed set.
  • Complemented lattice: a lattice with a unary operation, complementation, denoted by postfix ', such that x∧x' = 0 and 1'=0. 0 and 1 bound S.
    • Ortholattice: a complemented lattice such that x" = x and the complement is order reversing, so that x∨y=y ↔ y' ∨x' = x' .
  • Modular lattice: a lattice satisfying the modular identity, x∨(y∧(x∨z)) = (x∨y)∧(x∨z).
  • Distributive lattice: a lattice in which each of meet and join distributes over the other. Distributive lattices are modular, but the converse need not hold.
    • Kleene lattice: a bounded distributive lattice with a unary operation, denoted by postfix ', satisfying the axioms (x∨y)' = x'∨y', x" = x, and (x∨x')y∨y' = y∨y'.
    • Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.
      • Boolean algebra with operators: a Boolean algebra with one or more added operations. All added operations (a) evaluate to 0 if any argument is 0, and (b) are join preserving, i.e., distribute over join if binary. If postfix * denotes an additional unary operation, then 0* = 0 and (x∨y)* = x*∨y*.
        • Modal algebra: Boolean algebra with a single added unary operation, the modal operator.
          • Derivative algebra: a modal algebra whose added unary operation, the derivative operator, also satisfies x**∨x*∨x = x*∨x.
          • Interior algebra: a modal algebra whose added unary operation, the interior operator, also satisfies x*∨x = x and x** = x*. The dual is a closure algebra.
            • Monadic Boolean algebra: a closure algebra whose added unary operation, the existential quantifier, denoted by prefix ∃, satisfies the axiom ∃(∃x)' = (∃x)'. The dual operator, ∀x := (∃x' )' is the universal quantifier.

Three binary operations:

• Relation algebra: an interior algebra whose interior operator is called converse. S, the Cartesian square of some set, is a monoid under an added residuated binary operation, relative product, whose identity element is distinct from the Boolean bounds. Relative product distributes over join. The converse of a function is its inverse, and the relative product of two functions is their composition.

• Boolean semigroup: a Boolean algebra with an added binary operation that is associative, distributes over join, and is annihilated by 0.

• Implicative lattice: a distributive lattice with a third binary operation, implication, that distributes left and right over each of meet and join.
• Brouwerian algebra: a distributive lattice with a greatest element and a third binary operation, denoted by infix " ' ", satisfying ((x∧y)≤z)∧(y≤x)' z.

• Heyting algebra: a Brouwerian algebra with a least element, whose third binary operation, now called relative pseudo-complement, satisfies the identities x'x=1, x(x'y) = xy, x' (yz) = (x'y)(x'z), and (xy)'z = (x'z)(y'z).

Four binary operations:

• Residuated lattice: a Brouwerian algebra with a least element and an added binary operation, denoted by infix ⊗, such that (⊗,1) is a commutative monoid, obeying the adjointness property ((x≤y)' z) ↔ (x⊗y≤z).

[edit] Ring-like structures

Two binary operations, addition and multiplication. That multiplication has a 0 is either an axiom or a theorem.

• Shell: addition and multiplication have respective identity elements 0 and 1.
• Ringoid: multiplication distributes over addition..
  • Nonassociative ring: a ringoid that is an abelian group under addition.
    • Lie ring: a nonassociative ring whose multiplication anticommutes and satisfies the Jacobi identity.
    • Jordan ring: a nonassociative ring whose multiplication commutes and satisfies the Jordan identity.
  • Newman algebra: a ringoid that is also a shell. There is a unary operation, inverse, denoted by a postfix "'", such that x+x'=1 and xx'=0. The following are provable: inverse is unique, x"=x, addition commutes and associates, and multiplication commutes and is idempotent.
  • Semiring: a ringoid that is also a shell. Addition and multiplication associate, addition commutes.
    • Commutative semiring: a semiring whose multiplication commutes.
    • Kleene algebra: a semiring with idempotent addition and a unary operation, the Kleene star, denoted by postfix * and satisfying (1+x*x)x*=x*=(1+xx*)x*.
  • Rng: a ringoid that is an Abelian group under addition and 0, and a semigroup under multiplication.
    • Ring: a rng that is a monoid under multiplication and 1.
      • Commutative ring: a ring with commutative multiplication.
        • Boolean ring: a commutative ring with idempotent multiplication, having Boolean algebra as a model.

N.B. These definitions do not command universal assent:

• Some employ "ring" to denote what is here called a rng, and refer to a ring in the above sense as a "ring with identity";
• Some define a semiring as having no identity elements.

[edit] Modules and Algebras

Two sets, R and S. Elements of R are scalars, denoted by Greek letters. Elements of S are denoted by Latin letters. For every ring R, there is a corresponding variety of R-modules.

• Module: S is an abelian group with operators, each unary operator indexed by R. The operators are scalar multiplication RxS→S, which commutes, associates, is unital, and distributes over module and scalar addition. If only the pre(post)multiplication of module elements by scalars is defined, the result is a left (right) module.
  • Vector space: A module such that R is a field.
    • Group ring:
• Algebra over a ring (also R-algebra): a module where R is a commutative ring. There is a second binary operation over S, called multiplication and denoted by concatenation, which distributes over module addition and is bilinear: α(xy) = (αx)y = x(αy).
  • Algebra over a field: An algebra over a ring whose R is a field.
    • Associative algebra: an algebra over a field or ring, whose vector multiplication associates.
      • Commutative algebra: an associative algebra whose vector multiplication commutes.
      • Incidence algebra: an associative algebra such that the elements of S are the functions f [a,b]: [a,b]→R, where [a,b] is an arbitrary closed interval of a locally finite poset. Vector multiplication is defined as a convolution of functions.
      • Group algebra:
    • Jordan algebra: an algebra over a field whose vector multiplication commutes, may or may not associate, and satisfies the Jordan identity.
    • Lie algebra: an algebra over a field satisfying the Jacobi identity. The vector multiplication, the Lie bracket denoted [u,v], anticommutes, usually does not associate, and is nilpotent.

[edit] Structures that are not varieties

The structures in this section are not varieties because they cannot be axiomatized with identities alone. Most of the nonidentities are of three very simple kinds:

1. The requirement that S (or R or K) be a "nontrivial" ring, namely one such that S≠{0}, 0 being the additive identity element. The nearest thing to an identity implying S≠{0} is the nonidentity 0≠1, which requires that the additive and multiplicative identities be distinct.
2. Axioms involving multiplication, holding for all members of S (or R or K) except 0. In order for an algebraic structure to be a variety, the domain of each operation must be an entire underlying set; there can be no partial operations.
3. "0 is not the successor of anything," included in nearly all arithmetics.

Many mathematical structures of fundamental importance by virtue of their foundational nature (e.g., Peano arithmetic, the real field), or their richness (e.g., fields, normed vector spaces), are not varieties. Moreover, much of theoretical physics can be recast using multilinear algebras. Although structures with nonidentities retain an undoubted algebraic flavor, they suffer from defects varieties do not share. For example, neither the product of integral domains nor a free field over any set exists.

[edit] Arithmetics

One or both of the binary operations addition and multiplication. If both operations are included, the recursive identity defining multiplication links them. Arithmetics necessarily have infinite models.

• Cegielski arithmetic (Smorynski 1991): A commutative cancellative monoid under multiplication. 0 annihilates multiplication, and xy=1 if and only if x and y are both 1. Other axioms and one axiom schema govern order, exponentiation, divisibility, and primality; consult Smorynski. Adding the successor function and its axioms as per Dedekind algebra render addition recursively definable, resulting in a system with the expressive power of Robinson arithmetic.

In the structures below, addition and multiplication, if present, are recursively defined by means of successor, denoted by prefix σ. 0 is the axiomatic identity element for addition, and annihilates multiplication. Both axioms hold for semirings.

• Skolem arithmetic (Boolos and Jeffrey 2002: 73-76): Not an algebraic structure because there is no fixed set of operations of fixed adicity. Skolem arithmetic is a Dedekind algebra with projection functions, whose arguments are functions and that return any desired argument of a function. The identity function is the projection function whose arguments are all unary operations. Composite operations of any adicity, including addition and multiplication, may be constructed using function composition and primitive recursion. Induction becomes a theorem.
  • Kalmar arithmetic: Skolem arithmetic with different primitive functions.
  • Dedekind algebra (Potter 2004: 90): A pointed unary system whose operation, called successor, is injective. S is pointed and not a variety because it has a unique member, named 0, not included in the range of successor. All Dedekind algebras can be seen as fragments of Skolem arithmetic.
    • Dedekind-Peano structure: A Dedekind algebra with an axiom schema of induction.
      • Presburger arithmetic: A Dedekind-Peano structure with recursive addition.

Arithmetics above this line are decidable. Those below are incompletable.

• Robinson arithmetic: Presburger arithmetic with recursive multiplication.

• Primitive recursive arithmetic: Robinson arithmetic with possibly other recursively defined binary operations, and a rule of induction, stated schematically without quantifiers.
• Peano arithmetic: Robinson arithmetic with an axiom schema of induction. The semiring axioms for N (other than x+0=x and x0=0, included in the recursive definitions of addition and multiplication) are now theorems.

• Heyting arithmetic: Peano arithmetic with intuitionist logic as background logic.

[edit] Lattices that are not varieties

• Part algebra: a Boolean algebra lacking a least element 0, so that the complement of 1 is not defined.

[edit] Field-like structures

Two binary operations, addition and multiplication. S is nontrivial, i.e., S≠{0}.

• Domain: a ring whose sole zero divisor is 0.
  • Integral domain: a domain whose multiplication commutes. Also a commutative cancellative ring.
    • Euclidian domain: an integral domain with a function f: S→N satisfying the division with remainder property.
• Division ring (also sfield, skew field): a ring in which every member of S other than 0 has a two-sided multiplicative inverse. The nonzero members of S form a group under multiplication.
  • Field: a division ring whose multiplication commutes. Recapitulating: addition and multiplication commute, associate, and are unital. S is closed under a two-sided additive inverse, nonzero S under a two-sided multiplicative inverse. Multiplication distributes over addition.
    • Algebraically closed field: a field F such that all polynomial equations with coefficients in F also have roots in F.
    • Ordered field: a field whose elements are totally ordered. Sums and products of positive elements are positive.
      • Real field: a Dedekind complete ordered field.

The following structures are not varieties for reasons in addition to S≠{0}:

• Simple ring: a ring having no ideals other than 0 and S.
  • Weyl algebra:
• Artinian ring: a ring whose ideals satisfy the descending chain condition.

[edit] Vector spaces that are not varieties

The following composite structures are extensions of vector spaces that are not varieties. Two sets: M is a set of vectors and R is a set of scalars.

Three binary operations.

• Normed vector space: a vector space with a norm, namely a function M→R that is symmetric, linear, and positive definite.
  • Inner product space (also Euclidian vector space): a normed vector space such that R is the real field, whose norm is the square root of the inner product, M×M→R. Let i,j, and n be positive integers such that 1≤i,j≤n. Then M has an orthonormal basis such that e_i•e_j = 1 if i=j and 0 otherwise; see free module above.
  • Unitary space: Differs from inner product spaces in that R is the complex field, and the inner product has a different name, the hermitian inner product, with different properties: conjugate symmetric, bilinear, and positive definite. See Birkhoff and MacLane (1979: 369).
• Graded vector space: a vector space such that the members of M have a direct sum decomposition. See graded algebra below.

[edit] Multilinear algebras

Four binary operations. Two sets, V and K:

1. The members of V are multivectors (including vectors), denoted by lower case Latin letters. V is an abelian group under multivector addition, and a monoid under outer product. The outer product goes under various names, and is multilinear in principle but usually bilinear. The outer product defines the multivectors recursively starting from the vectors. Thus the members of V have a "degree" (see graded algebra below). Multivectors may have an inner product as well, denoted u•v: V×V→K, that is symmetric, linear, and positive definite; see inner product space above.
2. The properties and notation of K are the same as those of R above, except that K may have -1 as a distinguished member. K is usually the real field, as multilinear algebras are designed to describe physical phenomena without complex numbers.
3. The scalar multiplication of scalars and multivectors, V×K→V, has the same properties as module scalar multiplication.

• Symmetric algebra: a unital commutative algebra with vector multiplication.
• Universal enveloping algebra: Given a Lie algebra L over K, the "most general" unital associative K-algebra A, such that the Lie algebra A_L contains L.
  • Hopf algebra:
• Graded algebra: an associative algebra with unital outer product. The members of V have a direct sum decomposition resulting in their having a "degree," with vectors having degree 1. If u and v have degree i and j, respectively, the outer product of u and v is of degree i+j. V also has a distinguished member 0 for each possible degree. Hence all members of V having the same degree form an Abelian group under addition.
  • Tensor algebra: A graded algebra such that V includes all finite iterations of a binary operation over V, called the tensor product. All multilinear algebras can be seen as special cases of tensor algebra.
    • Exterior algebra (also Grassmann algebra): a graded algebra whose anticommutative outer product, denoted by infix ∧, is called the exterior product. V has an orthonormal basis. v₁ ∧ v₂ ∧ ... ∧ v_k = 0 if and only if v₁, ..., v_k are linearly dependent. Multivectors also have an inner product.
      • Clifford algebra: an exterior algebra with a symmetric bilinear form Q: V×V→K. The special case Q=0 yields an exterior algebra. The exterior product is written 〈u,v〉. Usually, 〈e_i,e_i〉 = -1 (usually) or 1 (otherwise).
      • Geometric algebra: an exterior algebra whose exterior (called geometric) product is denoted by concatenation. The geometric product of parallel multivectors commutes, that of orthogonal vectors anticommutes. The product of a scalar with a multivector commutes. vv yields a scalar.
        • Grassmann-Cayley algebra: a geometric algebra without an inner product.

[edit] Examples

Some recurring universes: N=natural numbers; Z=integers; Q=rational numbers; R=real numbers; C=complex numbers.

[edit] Arithmetics

• N is a pointed unary system, and the standard interpretation of Peano arithmetic.
• The universe of singletons forms a Dedekind-Peano structure if {x} interprets the successor of x, and the null set interprets 0 (Lewis 1991).

[edit] Group-like structures

• Nonzero N under addition is a magma.
• Z under addition (+) is an Abelian group.
• Z under subtraction (−) is a quasigroup.
• Nonzero Q under multiplication (×) is an Abelian group.
• Nonzero Q under division (÷) is a quasigroup.
• Every group is a loop, because a*x = b if and only if x = a⁻¹*b, and y*a = b if and only if y = b*a⁻¹.
• Invertible 2x2 matrices form a group under matrix multiplication.
• The permutations preserving the partition of a set induced by an equivalence relation form a group under function composition and inverse.
• Every cyclic group G is Abelian, because if x, y are in G, then xy = a^maⁿ = a^m+n = a^n+m = aⁿa^m = yx. In particular, Z is an Abelian group under addition, as are the integers modulo n, Z/nZ.
• The set of all functions X→X, X any nonempty set, is a monoid under function composition and the identity function.
• In category theory, the following are monoids under composition of morphisms and the identity morphism:
  • A category with a single object;
  • The set of all endomorphisms of an object X in category C .
• The Boolean algebra 2 is a boundary algebra.
• MV-algebras characterize multi-valued and fuzzy logics.
• More examples of groups and list of small groups.

Boolean algebras (BA) are primarily viewed as Boolean (complemented distributive) lattices. They are also ortholattices, rings, commutative monoids, and Newman algebras, and would be Abelian groups if the BA identity and inverse elements were identical instead of mutual complements.

[edit] Lattices

• The normal subgroups of a group, and the submodules of a module, form modular lattices.
• Any field of sets, and the connectives of first-order logic, are models of Boolean algebra. See Lindenbaum-Tarski algebra.
• The connectives of intuitionistic logic form a model of Heyting algebra.
• The modal logics K, S4, S5, and wK4 are models of modal algebra, interior algebra, monadic Boolean algebra, and derivative algebra, respectively.
• Peano arithmetic and most axiomatic set theories, including ZFC, NBG, and New Foundations, can be recast as models of relation algebra.

[edit] Ring-like structures

• N is a commutative semiring under addition and multiplication.
• The set R[X] of all polynomials over some coefficient ring R is a ring.
• 2x2 matrices under matrix addition and multiplication form a ring.
• If n is a positive integer, then the set Z_n = Z/nZ of integers modulo n (the additive cyclic group of order n ) forms a ring having n elements (see modular arithmetic).

[edit] Field-like structures

• Z is an integral domain under addition and multiplication.
• Each of Q, R, C, and the p-adic integers is a field under addition and multiplication.
• Q and R are ordered fields, totally ordered by "<" in the usual way
• R is the only Dedekind complete ordered field, as the axioms for such a field are categorical. The real field R grounds real and functional analysis.
  • R contains several interesting subfields, the algebraic, the computable, and the definable numbers.
• C is an algebraically closed field.
• An algebraic number field in number theory is a finite field extension of Q, that is, a field containing Q which has finite dimension as a vector space over Q.
• If q > 1 is a power of a prime number, then there exists (up to isomorphism) exactly one finite field with q elements, usually denoted F_q, or in the case that q is itself prime, by Z/qZ. Such fields are called Galois fields, whence the alternative notation GF(q). All finite fields are isomorphic to some Galois field.
  • Given some prime number p, the set Z_p = Z/pZ of integers modulo p is the finite field with p elements: F_p = {0, 1, ..., p − 1} where the operations are defined by performing the operation in Z, dividing by p and taking the remainder; see modular arithmetic.

[edit] Allowing additional structure

An algebraic structure can be endowed with additional structure, not necessarily algebraic in nature. The added structure must be compatible, in some sense, with the algebraic structure. Three kinds of additional structure are mentioned below: order, topology, and manifold.

S is an ordered set: These structures, all varieties, are discussed in Birkhoff (1967: chpts. 13-15, 17) using a differing terminology. Birkhoff also distinguishes partially ordered structures from the corresponding lattice-ordered ones.

• Ordered magma, semigroup, monoid, and group: S is partially ordered as well as having the requisite algebraic properties;
• Linearly ordered group: an ordered group such that the partial order on S is a linear order;
• Ordered ring: a ring whose S is linearly ordered;
• Archimedean group: a linearly ordered group for which the Archimedean property holds;
• Ordered vector space: a vector space such that M is partially ordered.

Topology:

• Topological group: a group whose S has a topology;
• Topological vector space: a normed vector space whose M has a topology.

Manifold:

• Lie group: a group whose S has a smooth manifold structure;

[edit] See also

• abstract algebra
• algebraic structure
• arity
• category theory
• free object
• universal algebra
• signature (universal algebra)
• variety (universal algebra)
• list of abstract algebra topics
• list of first order theories
• list of mathematics lists

[edit] References

• Garrett Birkhoff, 1967. Lattice Theory, 3rd ed, AMS Colloquium Publications Vol. 25. American Mathematical Society.
• --------, and Saunders MacLane, 1999 (1967). Algebra, 2nd ed. New York: Chelsea.
• George Boolos and Richard Jeffrey, 1980. Computability and Logic, 2nd ed. Cambridge Univ. Press.
• Dummit, David S., and Foote, Richard M., 2004. Abstract Algebra, 3rd ed. John Wiley and Sons.
• David K. Lewis, 1991. Part of Classes. Blackwell.
• Michel, Anthony N., and Herget, Charles J., 1993 (1981). Applied Algebra and Functional Analysis. Dover.
• Potter, Michael, 2004. Set Theory and its Philosophy, 2nd ed. Oxford Univ. Press.
• Smorynski, Craig, 1991. Logical Number Theory I. Springer-Verlag.

A monograph available free online:

• Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.

[edit] External links

• Jipsen's algebra structures. Includes many structures not mentioned here.
• Mathworld page on abstract algebra.