In this entry, all algebraic systems are of the same type. For each algebraic system, we drop the associated operator set for simplicity.

Let $A_i$ be algebraic systems indexed by $i\in I$ . $B$ is called a subdirect product of $A_i$ if

1. $B$ is a subalgebra of the direct product of $A_i$ .
2. for each $i\in I$ , $\pi_i(B)=A_i$ .

This generalizes the notion of a direct product, since in many instances, an algebraic system can not be decomposed into a direct product of algebras.

When all $A_i=C$ for some algebraic system $C$ of the same type, then $B$ is called a subdirect power of $C$ .

Remarks.

1. A very simple example of a subdirect product is the following: let $A_1=A_2=\lbrace 1,2,3\rbrace$ . Then the subset $B=\lbrace (x,y)\in A_1\times A_2 \mid x\le y \rbrace$ is a subdirect product of the sets $A_1$ and $A_2$ (considered as algebraic systems with no operators).
2. Let $B$ is a subdirect product of $A_i$ , and $p_i:=(\pi_i)_B$ , the restriction of $\pi_i$ to $B$ . Then $B/\ker(p_i)\cong A_i$ . In addition, $$\bigcap \lbrace \ker(p_i)\mid i\in I\rbrace=\Delta,$$ where $\Delta$ is the diagonal relation. To see the last equality, suppose $a,b\in B$ with $a\equiv b \pmod {p_i}$ . Then $a(i)=\pi_i(a)=p_i(a)=p_i(b)=\pi_i(b)=b(i)$ . Since this is true for every $i\in I$ , $a=b$ .
3. Conversely, if $A$ is an algebraic system and $\lbrace \mathfrak{C}_i\mid i\in I\rbrace$ is a set of congruences on $A$ such that $$\bigcap \lbrace \mathfrak{C}_i\mid i\in I\rbrace=\Delta.$$ Then $A$ is isomorphic to a subdirect product of $A/\mathfrak{C}_i$ .
4. An algebraic system is said to be subdirectly irreducible if, whenever $\mathfrak{C}_i$ are congruences on $A$ and $\bigcap \lbrace \mathfrak{C}_i\mid i\in I\rbrace=\Delta$ , then one of $\mathfrak{C}_i=\Delta$ .
5. Birkhoff's Theorem on the Decomposition of an Algebraic System. Every algebraic system is isomorphic to a subdirect product of subdirectly irreducible algebraic systems. This works only when the algebraic system is finitary.